Open Problems in Topology II

OPEN PROBLEMS IN TOPOLOGY II Edited By Elliott Pearl, Toronto, Canada Description This volume is a collection of survey...

Author: Elliott M. Pearl

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OPEN PROBLEMS IN TOPOLOGY II Edited By Elliott Pearl, Toronto, Canada Description This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis. Contents Part 1. General Topology 1. Selected ordered space problems (H. Bennett and D. Lutzer) 2. Problems on star-covering properties (M. Bonanzinga and M. Matveev) 3. Function space topologies (D.N. Georgiou, S.D. Iliadis and F. Mynard) 4. Spaces and mappings: special networks (C. Liu and Y. Tanaka) 5. Extension problems of real-valued continuous functions (H. Ohta and K. Yamazaki) 6. LE(k)-spaces (O. Okunev) 7. Problems on (ir) resolvability (O. Pavlov) 8. Topological games and Ramsey theory (M. Scheepers) 9. Selection principles and special sets of reals (B. Tsaban) Part 2. Set-theoretic Topology 10. Introduction: Twenty problems in set-theoretic topology (M. Hrusak and J.T. Moore) 11. Thin-tall spaces and cardinal sequences (J. Bagaria) 12. Sequential order (A. Dow) 13. On D-spaces (T. Eisworth) 14. The fourth head of BN (I. Farah) 15. Are stratifiable spaces M1? (G. Gruenhage) 16. Perfect compacta and basis problems in topology (G. Gruenhage and J.T. Moore) 17. Selection problems for hyperspaces (V. Gutev and T. Nogura) 18. Efimov's problem (K.P. Hart) 19. Completely separable MAD families (M. Hrusak and P. Simon) 20. Good, splendid and Jakovlev (I. Juhasz and W.A.R. Weiss) 21. Homogeneous compacta (J. van Mill) 22. Compact spaces with hereditarily normal squares (J.T. Moore) 23. The metrization problem for Frechet groups (J.T. Moore and S. Todorcevic) 24. Cech-Stone remainders of discrete spaces (P.J. Nyikos) 25. First countable, countably compact, noncompact spaces (P.J. Nyikos) 26. Linearly Lindelof problems (E. Pearl) 27. Small Dowker spaces (P.J. Szeptycki) 28. Reflection of topological properties to N1 (F.D. Tall) 29. The Scarborough-Stone problem (J.E. Vaughan) Part 3. Continuum Theory 30. Questions in and out of context (D.P. Bellamy) 31. An update on the elusive fixed-point property (C.L. Hagopian) 32. Hyperspaces of continua (A. Ilanes) 33. Inverse limits and dynamical systems (W.T. Ingram) 34. Indecomposable continua (W. Lewis) 35. Open problems on dendroids (V. Martinez-de-la-Vega and J.M. Martinez-Montejano) 36. ?Homogeneous continua (S.B. Nadler, Jr.) 37. Thirty open problems in the theory of homogeneous continua (J.R. Prajs) Part 4. Topological Algebra 38. Problems about the uniform structures of topological groups (A. Bouziad and J-P. Troallic) 39. On some special classes of continuous maps (M.M. Clementino and D. Hofmann) 40. Dense subgroups of compact groups (W.W. Comfort) 41. Selected topics from the structure theory of topological groups (D. Dikranjan and D. Shakhmatov) 42. Recent results and open questions relating Chu duality and Bohr compactifications of

locally compact groups (J. Galindo, S. Hernandez and T-S. Wu) 43. Topological transformation groups: selected topics (M. Megrelishvili) 44. Forty-plus annotated questions about large topological groups (V. Pestov) Part 5. Dynamical Systems 45. Minimal flows (W.F. Basener, K. Parwani and T. Wiandt) 46. The dynamics of tiling spaces (A. Clark) 47. Open problems in complex dynamics and "complex" topology (R.L. Devaney) 48. The topology and dynamics of flows (M.C. Sullivan) Part 6. Computer Science 49. Computational topology (D. Blackmore and T.J. Peters) Part 7. Functional Analysis 50. Non-smooth analysis, optimisation theory and Banach space theory (J.M. Borwein and W.B. Moors) 51. Topological structures of ordinary differential equations (V.V. Filippov) 52. The interplay between compact spaces and the Banach spaces of their continuous functions (P. Koszmider) 53. Tightness and t-equivalence (O. Okunev) 54. Topological problems in nonlinear and functional analysis (B. Ricceri) 55. Twenty questions on metacompactness in function spaces (V.V. Tkachuk) Part 8. Dimension Theory 56. Open problems in infinite-dimensional topology (T. Banakh, R. Cauty and M. Zarichnyi) 57. Classical dimension theory (V.A. Chatyrko) 58. Questions on weakly infinite-dimensional spaces (V.V. Fedorchuk) 59. Some problems in the dimension theory of compacta (B.A. Pasynkov) Part 9. Invited Papers 60. Problems from the Lviv topological seminar (T. Banakh, B. Bokalo, I. Guran, T. Radul and M. Zarichnyi) 61. Problems from the Bizerte-Sfax-Tunis Seminar (O. Echi, H. Marzougui and E. Salhi) 62. Cantor set problems (D.J. Garity and D. Repovs) 63. Problems from the Galway Topology Colloquium (C. Good, A. Marsh, A. McCluskey and B. McMaster) 64. The lattice of quasi-uniformities (E.P. de Jager amd H-P.A. Kunzi) 65. Topology in North Bay: some problems in continuum theory, dimension theory and selections (A. Karasev, M. Tuncali and V. Valov) 66. Moscow questions on topological algebra (K.L. Kozlov, E.A. Reznichenko and O.V. Sipacheva) 67. Some problems from George Mason University (J. Kulesza, R. Levy and M. Matveev) 68. Some problems on generalized metrizable spaces (S. Lin) 69. Problems from the Madrid Department of Geometry and Topology (J.M.R. Sanjurjo) 70. Cardinal sequences and universal spaces (L. Soukup) List of contributors Index Hardbound, 776 pages, publication date: MAR-2007 ISBN-13: 978-0-444-52208-5 ISBN-10: 0-444-52208-5

Preface This new book follows the 1990 volume Open Problems in Topology edited by J. van Mill and G.M. Reed. It builds on the success of original volume by presenting currently active research topics in topology. The contributions in this book are entirely new even though some (many?) of its problems may have been raised in other sources. As with the original volume, the intent is to provide a source of dissertation problems and to challenge the research community within and beyond topology. This volume covers a broad range of topics related to topology and examines some topics in greater depth. The problems in this volume are supposed to reflect the main trends in general topology and its applications since 1990 and I hope that they will help direct further research. This volume was prepared with the invaluable help of several editorial advisors. They are Frédéric Mynard (General Topology), Michael Hruˇsák and Justin Tatch Moore (Set-theoretic Topology), Alejandro Illanes (Continuum Theory), Dikran Dikranjan (Topological Algebra), John C. Mayer (Dynamical Systems), Thomas J. Peters (Computer Science), Biagio Ricceri (Functional Analysis), and Vitalij A. Chatyrko (Dimension Theory). The advisors helped with the selection of contributors and the review of manuscripts in the respective parts of the volume. This volume has a ninth part, Invited Problems, which is not organized by any particular topic. Here I invited several research groups to contribute problems representative of their interests. The journal Topology and its Applications will continue to publish regular reports on the status of problems from the original book and from this new volume. Please contribute information on solutions to the respective authors and editors. I want to thank Jan van Mill for his help in getting this project started. I thank the Fields Insititute in Toronto for their hospitality during the preparation of this volume. Elliott Pearl, Toronto, September 2006 v

Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved

Selected ordered space problems Harold Bennett and David Lutzer 1. Introduction A generalized ordered space (a GO-space) is a triple (X, τ, c when we replace “metric spaces” by “compact metric spaces”. Let X 2A space X is a Tanaka space if for a decreasing sequence {A : n ∈ N} with x ∈ A \ {x} n n for all n ∈ N, there exist xn ∈ An such that the sequence {xn : n ∈ N} converges to some point in X.

Products of k-spaces having certain k-networks

31

be a space with a point-countable k-network. When X 2 is a k-space, X is firstcountable or locally σ-compact (in view of [76]), thus χ(X) ≤ c. When X ω is a k-space, X is first-countable [30, 44]. Now, let X be a symmetric space having property (∗): any separable closed subset is a space whose points are Gδ -sets. A symmetric space has (∗) under CH, or if it is meta-Lindelöf or collectionwise Hausdorff. When X 2 is a k-space, χ(X) ≤ 2c . When X ω is a k-space, X is firstcountable. The authors don’t know whether a symmetric space Y which contains no (closed) copy of the space S2 is first-countable (here, Y is first-countable when Y has (∗)). If this is positive, then the above results hold without (∗). Problem 18.

107–108? 2

c

(1) Let X be a symmetric space. Is χ(X) ≤ c (or χ(X) ≤ 2 )? (2) Let X ω be a symmetric space. Is X first-countable? A space X has countable tightness if whenever x ∈ A, there is a countable subset C ⊂ A with x ∈ C. A space has countable tightness iff it has a determining cover by countable subsets [48]. A sequential space or a hereditarily separable space has countable tightness. For spaces X, Y having countable tightness, if X × Y is a k-space, then X × Y has countable tightness, and the converse holds when X, Y have a dominating cover by locally compact spaces. While, for a closed map f : X → Y with X strongly collectionwise Hausdorff, let Y 2 have countable tightness, then each boundary ∂f −1 (y) is c-compact (ω1 -compact if Y is sequential) [21]. Every product of spaces with a countable determining cover by locally separable metric subsets has countable tightness. But a product of a space with a point-finite determining cover (or a dominating cover) by ω1 many compact metric subsets may not have countable tightness. Liu and Lin [40] proved that the axiom b = ω1 is equivalent to the assertion that for k-spaces X, Y with a point-countable k-network by cosmic subsets (e.g., X, Y have a point-countable determining cover by locally separable metric subsets), X × Y has countable tightness iff one of X, Y is first-countable, or both X, Y are locally cosmic, and X 2 has countable tightness iff X is locally cosmic. They also showed that the axiom is equivalent to the assertion that for spaces X, Y with a dominating cover by metric subsets, if X × Y has a countable tightness, then one of X, Y is first-countable, or both X, Y have a countable dominating cover by metric subsets (equivalently, X, Y are ℵ-spaces). The converse of this assertion holds if the answer to (1) in the following problem is positive. Problem 19. (1) Let X be a space with a countable determining closed cover by metric subsets. Does X 2 have countable tightness? (2) Let X be a k-space with a point-countable k-network, in particular, let X be a (Fréchet) space with a point-countable (countable) determining closed cover by metric subsets. What is a necessary and sufficient condition for X 2 to have countable tightness?

109–110?

32

§4. Liu and Tanaka, Spaces and mappings: special networks

References [1] A. V. Arhangel0 ski˘ı, Mappings and spaces, Uspekhi Mat. Nauk 21 (1966), no. 4(130), 133– 184, Translation: Russian Math. Surveys 21 (1966) no. 4, 115–162. [2] D. A. Bonnett, A symmetrizable space that is not perfect, Proc. Amer. Math. Soc. 34 (1972), 560–564. [3] C. R. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1–16. [4] D. K. Burke and S. W. Davis, Questions on symmetrizable spaces, Questions Answers Gen. Topology 1 (1983), no. 2, 113–118. [5] H. Chen, Weak neighborhoods and Michael–Nagami’s question, Houston J. Math. 25 (1999), no. 2, 297–309. [6] H. Chen, Compact-covering maps and k-networks, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2623–2632. [7] G. D. Creede, Concerning semi-stratifiable spaces, Pacific J. Math. 32 (1970), 47–54. [8] S. W. Davis, G. Gruenhage, and P. J. Nyikos, Gδ sets in symmetrizable and related spaces, General Topology and Appl. 9 (1978), no. 3, 253–261. [9] J. Dugundji, Topology, Allyn and Bacon Inc., Boston, 1966. [10] R. Engelking, General topology, Heldermann Verlag, Berlin, 1989. [11] L. Foged, On g-metrizability, Pacific J. Math. 98 (1982), no. 2, 327–332. [12] L. Foged, A characterization of closed images of metric spaces, Proc. Amer. Math. Soc. 95 (1985), no. 3, 487–490. [13] L. Foged, Normality in k-and-ℵ spaces, Topology Appl. 22 (1986), no. 3, 223–240. [14] G. Gao, On k-networks and bases, J. Suzhou University 2 (1986), 107–111. [15] Z. M. Gao, A result on k-semistratifiable spaces, Questions Answers Gen. Topology 3 (1986), no. 2, 137–143. [16] Z. M. Gao, The closed images of metric spaces and Fr´ echet ℵ-spaces, Questions Answers Gen. Topology 5 (1987), no. 2, 281–291. [17] Z. M. Gao and Y. Hattori, A characterization of closed s-images of metric spaces, Questions Answers Gen. Topology 4 (1987), no. 2, 147–151. [18] G. Gruenhage, k-spaces and products of closed images of metric spaces, Proc. Amer. Math. Soc. 80 (1980), no. 3, 478–482. [19] G. Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 423–501. [20] G. Gruenhage, E. Michael, and Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984), no. 2, 303–332. [21] G. Gruenhage and Y. Tanaka, Products of k-spaces and spaces of countable tightness, Trans. Amer. Math. Soc. 273 (1982), no. 1, 299–308. [22] R. W. Heath, On spaces with point-countable bases, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys. 13 (1965), 393–395. [23] T. Hoshina, On the quotient s-images of metric spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 10 (1970), 265–268 (1970). [24] N. N. Jakovlev, On the theory of o-metrizable spaces, Dokl. Akad. Nauk SSSR 229 (1976), no. 6, 1330–1331, Translation: Soviet Math. Dokl. 17 (1976) 1217–1219. [25] H. Junnila and Z. Q. Yun, ℵ-spaces and spaces with σ-hereditarily closure-preserving knetwork, Topology Appl. 44 (1992), 209–215. [26] N. Laˇsnev, Closed images of metric spaces, Dokl. Akad. Nauk SSSR 170 (1966), 505–507, Translation: Soviet Math. Dokl. 7 (1966) 1219–1221. [27] S. Lin, Mapping theorems on ℵ-spaces, Topology Appl. 30 (1988), no. 2, 159–164. [28] S. Lin, On g-metrizable spaces, Chinese Ann. Math. Ser. A 13 (1992), no. 3, 403–409. [29] S. Lin, Generalized metric spaces and mappings, Chinese Science Press, Beijing, 1995. [30] S. Lin, A note on the Arens’ space and sequential fan, Topology Appl. 81 (1997), no. 3, 185–196.

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[61] M. Sakai, Counterexamples on generalized metric spaces, Sci. Math. Jpn. 64 (2006), no. 1, 73–76. [62] A. Shibakov, Sequentiality of products of spaces with point-countable k-networks, Topology Proc. 20 (1995), 251–270. [63] A. Shibakov, Metrizability of sequential topological groups with point-countable k-networks, Proc. Amer. Math. Soc. 126 (1998), no. 3, 943–947. [64] R. Sirois-Dumais, Quasi- and weakly-quasi-first-countable spaces, Topology Appl. 11 (1980), no. 2, 223–230. [65] F. Siwiec, On defining a space by a weak base, Pacific J. Math. 52 (1974), 233–245. [66] R. M. Stephenson, Jr., Symmetrizable, F-, and weakly first countable spaces, Canad. J. Math. 29 (1977), no. 3, 480–488. [67] S. A. Svetlichny˘ı, Intersection of topologies and metrizability in topological groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1989), no. 4, 79–81, Translation: Moscow Univ. Math. Bull. 44 (1989) no. 4, 78–80. [68] Y. Tanaka, On local properties of topological spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 11 (1972), 106–116. [69] Y. Tanaka, A characterization for the products of k- and ℵ0 -spaces and related results, Proc. Amer. Math. Soc. 59 (1976), no. 1, 149–155. [70] Y. Tanaka, Closed maps on metric spaces, Topology Appl. 11 (1980), no. 1, 87–92. [71] Y. Tanaka, Point-countable covers and k-networks, Topology Proc. 12 (1987), no. 2, 327– 349. [72] Y. Tanaka, Metrization II, Topics in general topology, North-Holland, Amsterdam, 1989, pp. 275–314. [73] Y. Tanaka, σ-hereditarily closure-preserving k-networks and g-metrizability, Proc. Amer. Math. Soc. 112 (1991), no. 1, 283–290. [74] Y. Tanaka, Closed maps and symmetric spaces, Questions Answers Gen. Topology 11 (1993), no. 2, 215–233. [75] Y. Tanaka, Theory of k-networks, Questions Answers Gen. Topology 12 (1994), no. 2, 139– 164. [76] Y. Tanaka, Products of k-spaces having point-countable k-networks, Topology Proc. 22 (1997), Spring, 305–329. [77] Y. Tanaka, Theory of k-networks. II, Questions Answers Gen. Topology 19 (2001), no. 1, 27–46. [78] Y. Tanaka, Products of k-spaces, and questions, Comment. Math. Univ. Carolin. 44 (2003), no. 2, 335–345. [79] Y. Tanaka, Quotient spaces and decompositions, Encyclopedia of general topology (K. P. Hart, J. Nagata, and J. E. Vaughan, eds.), Elsevier Science, 2004, pp. 43–46. [80] Y. Tanaka, Products of weak topologies, Topology Proc. 29 (2005), no. 1, 361–376. [81] Y. Tanaka, Determining covers and dominating covers, Questions Answers Gen. Topology 24 (2006), no. 2.


Extension problems of real-valued continuous functions Haruto Ohta and Kaori Yamazaki 1. Introduction By a space we mean a completely regular T1 -space. A subset A of a space X is said to be C-embedded in X if every real-valued continuous function on A extends continuously over X, and A is said to be C ∗ -embedded in X if every bounded real-valued continuous function on A extends continuously over X. The aim of this paper is to collect some open questions concerning C-, C ∗ -embeddings and extension properties which can be described by extensions of real-valued continuous functions. Let N, Q, R and I denote the sets of natural numbers, rationals, reals, and the closed unit interval, respectively, with the usual topologies. Let ω be the first infinite cardinal. For undefined terms on generalized metric spaces, see [9]. General terminology and notation will be used as in [6]. 2. C-embedding versus C ∗ -embedding This section overlaps partly with the survey [28]; here, we update information about status of the questions and add some new questions. It is not difficult to construct an example of a closed set which is C ∗ -embedded but not C-embedded (for example, see [26, Construction 2.3]). It is, however, interesting to ask if C ∗ -embedding implies C-embedding under certain circumstances. Question 1. Is every C ∗ -embedded subset of a first countable space C-embedded? Note that every C ∗ -embedded subset of a first countable space is closed. Kulesza–Levy–Nyikos [18] proved that if b = s = c, then there exists a maximal almost disjoint family R of infinite subsets of N such that every countable set of nonisolated points of the space N∪R is C ∗ -embedded. Since every set of nonisolated points of N ∪ R is discrete and N ∪ R is pseudocompact, those countable sets are not C-embedded. Thus, Question 1 has a negative answer under b = s = c, but it remains open whether there exists a counterexample in ZFC. Kulesza–Levy– Nyikos [18] also proved that, assuming the Product Measure Extension Axiom (PMEA), there exists no infinite discrete C ∗ -embedded subset of a pseudocompact space of character less than c. Since every C ∗ - but not C-embedded subset contains an infinite discrete C ∗ -embedded subset, this implies that no pseudocompact space, in particular, no space N ∪ R, can be a counterexample to Question 1 under PMEA (see [17] for related results). On the other hand, by Tietze’s extension Research of the second author is partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), No. 16740028. 35

111?

36

§5. Ohta and Yamazaki, Extension problems of real-valued continuous functions

theorem, no normal space can also be a counterexample to Question 1. Thus, it is natural to ask if typical examples of first countable, non-normal spaces contain C ∗ -embedded subsets which are not C-embedded. In [28] the first author proved that every C ∗ -embedded subset of the Niemytzki plane NP is C-embedded and asked the following questions, which have been unanswered as yet. 112?

113?

Question 2. Is every C ∗ -embedded subset of the square S 2 of the Sorgenfrey line C-embedded in S 2 ? Question 3. Is every C ∗ -embedded subset of the product RQ × Nω of the Michael line with the space of irrationals C-embedded in RQ × Nω ? It was proved in [28] that if a space X contains a pair of disjoint closed sets, one of which is countable discrete, which cannot be separated by disjoint open sets, then the absolute E(X) of X contains a closed C ∗ -embedded subset which is not C-embedded. Hence, the absolutes E(NP), E(S 2 ) and E(RQ × Nω ) of the Niemytzki plane, the Sorgenfrey plane and Michael’s product space, respectively, contain C ∗ - but not C-embedded closed sets. Another interesting case of the relationship between C ∗ - and C-embeddings is a closed rectangle in a product. Indeed, the next question asked in [28] and the one after next are still open.

114?

Question 4. Let A be a C-embedded closed subset of a space X and Y a space such that A × Y is C ∗ -embedded in X × Y . Then, is A × Y C-embedded in X × Y ?

115?

Question 5. Let A (resp. B) be a C-embedded closed subset of a space X (resp. Y ) such that A × B is C ∗ -embedded in X × Y . Then, is A × B C-embedded in X × Y ? It is known that the answer to Question 4 (resp. 5) is positive in each of the following cases (1)–(4) (resp. (5) and (6)): (1) Y is the product of a σ-locally compact (i.e., the union of countably many locally compact closed subspaces), paracompact space with a metric space ([28, Corollary 4.10]). In particular, Y is either σ-locally compact, paracompact ([43, Theorem 1.1]) or a metric space ([11, Theorem 1.1]). (2) Y ≈ Y 2 and Y contains an infinite compact set ([16, Theorem 2.1]). (3) X is a normal P -space and Y is a paracompact Σ-space ([43, Theorem 1.2]). (4) X is a normal weak P (ω)-space and Y is K-analytic ([48, Theorem 4.2]). (5) Y is locally compact and paracompact (combine [21, Theorem 4] with [43, Theorem 1.1]). (6) Y is a metric space (combine [11, Theorem 1.1] with [38, Theorem 4]). Question 4 is a special case of Question 5, and the authors do not know if the answer to Question 5 is positive in each of the cases (1)–(4) above. As for another candidate for positive cases, the following question naturally arises; the case where Y is a paracompact M -space was asked by Gutev and the first author in [11, Problem 1].

C-embedding versus C ∗ -embedding

37

Question 6. Is an answer to Question 4 positive if either the space X or Y is assumed to be a paracompact M -space (or equivalently, a paracompact p-space)?

116?

Recall from [13] that a subset A of a space X is said to be U ω -embedded in X if for every real-valued continuous function g on A there exists a real-valued continuous function f on X such that g(x) ≤ f (x) for each x ∈ A (see [11, Lemma 2.5]). It is known that A is C-embedded in X if and only if A is C ∗ - and U ω -embedded in X, i.e., C = C ∗ + U ω . Gutev and the first author [11] proved that if A is a C-embedded subset of a space X and Y is a metric space, then A × Y is C ∗ -embedded in X × Y if and only if A × Y is U ω -embedded in X × Y . Neither the ‘if’ part nor the ‘only if’ part of this result is known to be true for any generalized metric spaces Y . In particular, the following question is open (see [11, Problems 1 and 2] for related questions). Question 7. Let A be a C-embedded subset of a space X and Y a stratifiable space.

117–118?

(i) Is A × Y C ∗ -embedded in X × Y provided that A × Y is U ω -embedded in X ×Y? (ii) Is A × Y U ω -embedded in X × Y provided that A × Y is C ∗ -embedded in X ×Y? Next, we turn to questions on infinite products. For an infinite cardinal γ, let us consider the following condition p(γ). p(γ): For every collection of pairs (Xα , Aα ), α < γ, ofQa space Xα and its closed subset Aα with |Aα | > 1, if the product A = α 1?

592?

Problem 9 ([26, Question 4.12]). Does there exist a continumm X that is not an arc, for which there is an integer n > 1 such that Cn (X) is homeomorphic to the product of two finite-dimensional continua?

593?

In this area, it would be interesting to characterize those continua X for which Fn (X) is homeomorphic to the cone over some continuum Z and it would be also interesting to characterize those continua X for which Fn (X) is homeomorphic to the product of two nondegenerate continua. There are only a few results on this direction. E. Casta˜ neda showed in [4] that: (a) if X is a finite graph, then F2 (X) is the cone over a continuum Y if and only if X is a simple m-od or an arc and, (b) if X is a finite graph, then F2 (X) is a product of two nondegenerate continua if and only if X is an arc. On the other hand, it is known (see [1, Theorem 6]) that F3 ([0, 1]) is homeomorphic to [0, 1]3 . In [3, Lemma 1], E. Casta˜ neda proved that If X is a simple m-od, then F2 (X) is homeomorphic to the cone over a continuum Z, this result was extended in [25] for every Fn (X). The following problems are open. Problem 10 ([4, Question 3.15]). Is [0, 1] the only finite graph such that F3 (X) is a product of two nondegenerate continua?

594?

Problem 11. Do there exist a finite graph X and an integer n ≥ 4 such that Fn (X) is a product of two nondegenerate continua?

595?

Problem 12. Are the simple m-ods and the arc the only finite graphs for which Fn (X) (n ≥ 3) is the cone over a continuum Y ? This problem is interesting even for n = 3.

596?

Problem 13 ([25, Question 3.5]). Does there exist a continuum X which is not the cone over a compactum such that Fn (X) is homeomorphic to the cone over a finite-dimensional continuum for some integer n ≥ 2?

597?

Means A mean is a continuous function m : F2 (X) → X such that m({x}) = x for each x ∈ X. The main problem in this area is to charaterize those continua X which admit a mean. Many authors have studied this problem and there are a number of open problems on this area. In this section we only include some of

282

§32. Illanes, Hyperspaces of continua

my favorite questions, all of them appeared in [20], where the following results were obtained. (a) Each dendrite admits a monotone mean, while the harmonic fan admits no monotone mean. (b) Each n-cell, as well as the dyadic solenoid, admits a mean that is monotone and open, simultaneously. (c) Each simple n-od, as well as the Cantor fan admits no open mean. (d) The harmonic fan admits a confluent mean. The interested reader can find more information and problems about means in [19, Ch. XII, Section 76] and [5]. 598?

Problem 14 ([20, Question 2.3]). Suppose that X is a dendroid and it admits a monotone mean, does it follow that X is a dendrite?

599?

Problem 15 ([20, Question 3.8]). Does each tree admit an open mean?

600?

Problem 16 ([20, Question 3.9]). Does there exist a dendrite X such that X is not a tree and X admits and open mean?

601?

Problem 17 ([20, Question 4.2]). Does there exist a continuum X such that X admits a mean but X does not admit a confluent mean? Fixed point property B. Knaster asked in The Scottish book, in 1952, whether C(X) must have the fixed point property when X has the fixed point property. In [32] S.B. Nadler, Jr. and J.T. Rogers, Jr. showed that if Y is the union of a disk D with a ray surrounding the boundary of D, then Y the fixed point property but C(Y ) and 2Y do not have the fixed point property. So, J.T. Rogers asked if C(X) has the fixed point when X is a tree-like continuum [28, Problem 446, p. 307]. Recently, the author has answered this question in [9] by showing that if Z is the union of a simple triod T with a ray surrounding it, then C(Z) does not have the fixed point property. It is not known if the statements (a) C(X) has the fixed point property and, (b) 2X has the fixed point property; are equivalent [29, Question 7.12, p. 299].

602?

Problem 18. Let Z be as in the previous paragraph. Does 2Z have the fixed point property? With respect to symmetric products, in the paper in which they were introduced [1], K. Borsuk and S. Ulam asked whether Fn (X) has the fixed point property when X has the fixed point property. This question was solved by J. Oledzki in 1988 [33], who gave an example of a 2-dimensional continuum X with the fixed point property such that F2 (X) does not have the fixed point property. With this in mind, S.B. Nadler, Jr. has offered the following list of new questions (among others) in [31].

603?

Problem 19 ([31, 8.12, p. 119]). Does Fn (X) have the fixed point property when X is a circle-like continuum with the fixed point property?

Mappings between hyperspaces

283

Problem 20 ([31, 8.13, p. 119]). Does Fn (X), n ≥ 2, have the fixed point property when X is hereditarily indecomposable continuum with the fixed point property?

604?

The answer to Problem 20 is not known even when X is the pseudo-arc and n ≥ 3. Problem 21 ([31, 8.14, p. 120]). Is there a 1-dimensional continuum X with the fixed point property such that Fn (X) does not have the fixed point property for some n?

605?

Problem 22 ([31, 8.15, p. 120]). If X is a tree-like continuum with the fixed point property then does Fn (X) have the fixed point property?

606?

Problem 23 ([31, 8.16, p. 120]). Is there a continuum X such that Fn (X) has the fixed point property for some n > 1 and, yet, Fm (X) does not have the fixed point property for some m?

607?

Problem 24 ([31, 8.17, p. 121]). Is there a continuum X such that X and F2 (X) have the fixed point property but F3 (X) does not have the fixed point property?

608?

Problem 25 ([31, 8.23, p. 122]). If X is a continuum such that C(X) has the fixed point property, then does Cn (X) have the fixed point property for each n?

609?

Problem 26 ([31, 8.24, p. 123]). If X is a continuum such that Fn (X) has the fixed point property for all n, then does Cn (X) have the fixed point property for all n?

610?

Let Z be as in the first paragraph at the beginning of this section (a simple triod with a ray surrounding it). Since it has been proved [9] that C(Z) does not have the fixed point property, it would be interesting to know if Fn (Z) has the fixed point property for each n. If this is true, then Z would provide a negative answer to Problem 26. If this is false, then Problem 22 would be solved in the negative. Problem 27. Let Z be as in the previous paragraph, does Fn (Z) have the fixed point property for all n?

611?

Problem 28 ([31, 8.26, p. 123]). If X is a continuum such that Fn (X) has the fixed point property for all n, then does 2X have the fixed point property?

612?

Problem 29 ([31, p. 77]). Does Fn (X) have the fixed point property when X is arc-like and n ≥ 3?

613?

It is known that if X is arc-like, then F2 (X) has the fixed point property ([19, 22.25, p. 199]). Mappings between hyperspaces In [29, 22.25, p. 199], S.B. Nadler, Jr., discussed the problem of when there exists a continuous map from one of the continua C(X), 2X or X onto another. On this topic, I. Krzemi´ nska and J.R. Prajs [21] have shown that there exists a uniformly path connected continuum X such that X is not a continuous image of C(X). Recently, A. Illanes [12], has constructed a continuum X such that C(X) is not a continuous image of X. The following questions remain open.

284


614?

Problem 30 ([21, 22.25, p. 199]). If f : X → Y is a continuous surjection between continua, does there exist a continuous surjection g : C(X) → C(Y )?

615?

Problem 31 ([21, Question 3, p. 61]). Given a continuum X, does there exist a continuous surjection f : C(X) → C(C(X))? Unicoherence of F2 (S 1 ) Answering a question by A. Illanes and A. Garc´ıa-Máynez, E. Casta˜ neda showed (in [2]) that if S1 and S2 are the circles in the plane, centered at the origin, with radius 1 and 2, respectively and, R is a topological copy of the real line such that one end surrounds asymptotically the circle S1 and the other end surrounds S2 , then X = S1 ∪ S2 ∪ R is unicoherent but F2 (X) is not unicoherent. It is known that if X is a locally connected unicoherent continuum, then F2 (X) is unicoherent [10]. A discussion on this topic can be found in [7]. The following questions are open.

616?

Problem 32 ([2, Problem 1, p. 66]). Does there exist an indecomposable continuum X such that F2 (X) is not unicoherent?

617?

Problem 33 ([2, Problem 2, p. 66]). Does there exist a hereditarily unicoherent continuum X such that F2 (X) is not unicoherent?

618?

Problem 34 (J.J. Charatonik, [2, Problem 3, p. 66]). Does there exist a hereditarily unicoherent, hereditarily decomposable continuum X such that F2 (X) is not unicoherent? Locating cells in hyperspaces In [23], S. L´ opez made a very detailed study of those continua X for which the element X in C(X) has a neighborhood in C(X) which is a 2-cell. The following question remains open.

619?

Problem 35 ([23, Question 10, p. 189]). Suppose that there is a neighborhood D of X in C(X) such that D is embeddable in the plane. Does X have a neighborhood in C(X) which is a 2-cell? Locating m-cells in hyperspaces has been an important tool in the study of hyperspaces. An m-od in a continuum X is a subcontinuum B for which there exists a subcontinuum A ⊂ B such that B − A contains at least m components. When C1 , . . . , Cm are components of B − A, taking an order arc αi from A to A ∪ clX (Ci ), for each i ∈ {1, . . . , m} (that is, αi : [0, 1] → C(X) is a continuous function such that αi (0) = A, αi (1) = A ∪ clX (Ci ) and αi (s) ( αi (t) if s < t, for the existence of order arcs see [19, Theorem 15.3]) and defining ϕ : [0, 1]m → C(X) by ϕ((t1 , . . . , tm )) = α1 (t1 ) ∪ · · · ∪ αm (tm ), it is easy to see that ϕ is an embedding. Thus, if there exists an m-od in X, then there exists an m-cell in C(X). The converse of this implication is also true (see [19, Theorem 70.1]). Therefore, we have a complete intrinsic charaterization of those continua X for which there exists an m-cell in C(X). It would be interesting to have a similar characterization for

Unique hyperspaces

285

the hyperspaces Cn (X). With the idea described above it can be proved that if B1 , . . . , Bn are pairwise disjoint subcontinua of X such that each Bi is an ri -od, then there exists a (r1 +· · ·+rn )-cell in Cn (X). The problem here is to determine if this is the only way to obtain cells in Cn (X). Thus we have the following problem. Problem 36. Suppose that X is a continuum such that there exists an m-cell in Cn (X), then does there exist pairwise disjoint subcontinua B1 , . . . , Bn of X such that each Bi is an ri -od and m = r1 + · · · + rn ?

620?

Unique hyperspaces The continuum X is said to have unique hyperspace C(X) (2X , Cn (X) and Fn (X), respectively) provided that if Y is a continuum and C(X) (2X , Cn (X) and Fn (X), respectively) is homeomorphic to C(Y ) (2Y , Cn (Y ) and Fn (Y ), respectively), then X is homeomorphic to Y . A discussion on what is known about unique hyperspaces can be found in [13]. The following questions are open. Problem 37 ([13, p. 77]). Do hereditarily indecomposable continua X have unique hyperspace F2 (X)?

621?

Problem 38 ([8, p. 93]). Let X and Y be dendrites whose respective sets of endpoints are closed. Suppose that C2 (X) is homeomorphic to C2 (Y ), then does it follow that X is homeomorphic to Y ?

622?

The respective question for Cn (X) instead of C2 (X) with n 6= 2 has been solved in the affirmative in [8, Theorem 5.24]. Problem 39. Let X be a dendrite and let Y be a continuum such that Fn (X) is homeomorphic to Fn (Y ), for some n ≥ 3. Does it follow that Y is also a dendrite?

623?

The respective question for n = 2 was answered in the positive in [13]. Problem 40. Let X and Y be a dendrites. Suppose that the respective sets of ordinary points (that is, sets of non-ramification points) are open and Fn (X) is homeomorphic to Fn (Y ), for some n ≥ 3. Does it follow that X is homeomorphic to Y ?

624?

The respective question for n = 2 was answered in the positive in [13]. Problem 41. Let X and Y be metric compactifications of the ray [0, 1). Suppose that F3 (X) and F3 (Y ) are homeomorphic. Does it follow that X and Y are homeomorphic?

625?

The respective question for n 6= 3 has been recently solved in the affirmative by J.M. Mart´ınez-Montejano. Problem 42. Let X be a metric compactification of the ray [0, 1) and let Y be a continuum. Suppose that Fn (X) is homeomorphic to Fn (Y ) for some n > 1. Does it follow that Y is a metric compactification of the ray [0, 1)?

626?

286


1. Miscellaneous problems In [3], E. Casta˜ neda, showed that if X is a locally connected continuum, then F2 (X) can be embedded in R3 if and only if X can be embedded in the figure-8 curve (the continuum obtained by joining two simple closed curves by a point). The following problem is open. 627?

Problem 43. Can F2 (X) be embedded in R4 for each finite graph X? A topological property P is said to be sequential decreasing Whitney property provided that if µ is a Whitney map for C(X), {tn }∞ n=1 is a sequence in the interval (t, µ(X)) such that lim tn = t and each Whitney level µ−1 (tn ) has property P, then µ−1 (t) has property P. Sequential decreasing Whitney properties were introduced and studied by F. Orozco-Zitli in [34], where he posed the following problem.

628?

Problem 44 ([34, Question 1, p. 305]). Is the property of being a hereditarily arcwise connected continuum a sequential decreasing Whitney property? Let As (X) = {A ∈ C(X) : A is an arc} ∪ F1 (X). It is known that if X is a dendrite, then As (X) is homeomorphic to F2 (X), by associating each arc with its set of end-points. On the other hand, in [15], it has been proved that if X is a dendroid with only one ramification point and F2 (X) is homeomorphic to As (X), then X is a dendrite. So the following question arises naturally.

629?

Problem 45 (A. Soto, [15, Question 1, p. 308]). If X is a dendroid such that F2 (X) and As (X) are homeomorphic, must X be a dendrite? Given a continuous function between continua f : X → Y the induced mapping 2f : 2X → 2Y is defined by 2f (A) = f (A) (the image of A under f ). A wide discussion on what has been done about induced mappings can be found in [19, Ch. XII, Section 77].

630?

Problem 46. Suppose that X is hereditarily indecomposable and F : 2X → 2X is a homeomorphism, is it true that there exist a homeomorphism f : X → X such that F = 2f ? The continuum X is said to be zero-dimensional closed set aposyndetic provided that for each zero-dimensional closed subset A of X and for each p ∈ X − A, there exists a subcontinuum M of X such that p ∈ int(M ) and M ∩ A = ∅. Answering a question by J. Goodykoontz, Jr., recently, J.M. Mart´ınez-Montejano has shown that the hyperspaces 2X and Cn (X) (for all n) are zero-dimensional closed set aposyndetic and he offered the following question.

631?

Problem 47 ([24, Question 3.1]). Let n ≥ 3. Is Fn (X) zero-dimensional closed set aposyndetic?

632?

Problem 48 ([16, Problem 1, p. 180]). Are there integers 1 < n < m and continua X and Y such that dim C(X) is finite and Cn (X) is homeomorphic to Cm (Y )? Some partial answers to this question are given in [16].

References

287

Problem 49 ([16, Problem 2, p. 180]). Do there exist two non-homeomorphic continua X and Y such that Cn (X) is homeomorphic to Cn (Y ), dim Cn (X) is finite and n > 1?

633?

Given p ∈ X, in [36], the following map was considered: ϕp : X → F2 (X), given by ϕp (x) = {p, x}. In the same paper, P. Pellicer-Covarrubias proved that [36, Lemma 5.3] if X is contractible, then ϕp is a deformation retraction in F2 (X). This motivates the following problem. Problem 50 ([36, p. 291]). Suppose that ϕp is a deformation retraction, does this imply that X is contractible?

634?

Given p ∈ X, let C(p, X) = {A ∈ C(X) : p ∈ A} and let K(X) = {C(p, X) ∈ C(C(X)) : p ∈ X}. Spaces of the form K(X) were studied by P. PellicerCovarrubias in [35], where she posed the following problems. Problem 51 ([35, p. 284]). Let T be a simple triod. Does there exists a continuum X such that X is not homeomorphic to T and K(X) is homeomorphic to K(T )? If so, must X be indecomposable?

635?

Problem 52 ([35, p. 284]). Let G be a finite graph. Does there exists a continuum X such that X is not homeomorphic to G and K(X) is homeomorphic to K(G)? If so, must X be indecomposable?

636?

We finish this list of problems including an interesting problem of Continuum Theory (not of hyperspaces) which has not been posed in the literature. For the particular case when X is the pseudo-arc, this question was solved in [27]. Problem 53. Is it true that for each continuum X there exists an uncountable family of pairwise non-homeomorphic metric compactifications of the ray [0, 1) with remainder X. References [1] K. Borsuk and S. Ulam, On symmetric products of topological spaces, Bull. Amer. Math. Soc. 37 (1931), 875–882. [2] E. Casta˜ neda, A unicoherent continuum whose second symmetric product is not unicoherent, Topology Proc. 23 (1998), Spring, 61–67. [3] E. Casta˜ neda, Embedding symmetric products in Euclidean spaces, Continuum theory (Denton, TX, 1999), Lecture Notes in Pure and Appl. Math., vol. 230, Marcel Dekker Inc., New York, 2002, pp. 67–79. [4] E. Casta˜ neda, Symmetric products as cones and products, Topology Proc. 28 (2004), no. 1, 55–67. [5] J. J. Charatonik, Some problems concerning means on topological spaces, Topology, measures, and fractals (Warnem¨ unde, 1991), Akademie-Verlag, Berlin, 1992, pp. 166–177. [6] W. J. Charatonik and A. Samulewicz, On size mappings, Rocky Mountain J. Math. 32 (2002), no. 1, 45–69. [7] A. Garc´ıa-M´ aynez and A. Illanes, A survey on unicoherence and related properties, An. Inst. Mat. Univ. Nac. Aut´ onoma M´ exico 29 (1989), 17–67 (1990). [8] D. Herrera-Carrasco, Hiperespacios de dendritas, Doctoral dissertation, Facultad de Ciencias, Universidad Nacional Aut´ onoma de M´ exico, November, 2005.

637?

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[9] A. Illanes, A tree-like continuum whose hyperspace of subcontinua has a fixed-point-free map, Preprint. [10] A. Illanes, Multicoherence of symmetric products, An. Inst. Mat. Univ. Nac. Aut´ onoma M´ exico 25 (1985), 11–24. [11] A. Illanes, Hyperspaces which are products, Topology Appl. 79 (1997), no. 3, 229–247. [12] A. Illanes, A continuum whose hyperspace of subcontinua is not g-contractible, Proc. Amer. Math. Soc. 130 (2002), no. 7, 2179–2182. [13] A. Illanes, Dendrites with unique hyperspace F2 (X), JP J. Geom. Topol. 2 (2002), no. 1, 75–96. [14] A. Illanes, The hyperspace C2 (X) for a finite graph X is unique, Glas. Mat. Ser. III 37(57) (2002), no. 2, 347–363. [15] A. Illanes, Hyperspaces of arcs and two-point sets in dendroids, Topology Appl. 117 (2002), no. 3, 307–317. [16] A. Illanes, Comparing n-fold and m-fold hyperspaces, Topology Appl. 133 (2003), no. 3, 179–198. [17] A. Illanes, A model for the hyperspace C2 (S 1 ), Questions Answers Gen. Topology 22 (2004), no. 2, 117–130. [18] A. Illanes and M. de J. L´ opez, Hyperspaces homeomorphic to cones. II, Topology Appl. 126 (2002), no. 3, 377–391. [19] A. Illanes and S. B. Nadler, Jr., Hyperspaces, Marcel Dekker Inc., New York, 1999. [20] A. Illanes and L. C. Sim´ on, Means with special properties, Houston J. Math. 29 (2003), no. 2, 313–324. [21] I. Krzemi´ nska and J. R. Prajs, On continua whose hyperspace of subcontinua is σ-locally connected, Topology Appl. 96 (1999), no. 1, 53–61. [22] M. de J. L´ opez, Hyperspaces homeomorphic to cones, Topology Appl. 126 (2002), no. 3, 361–375. [23] S. L´ opez, Hyperspaces locally 2-cell at the top, Continuum theory (Denton, TX, 1999), Lecture Notes in Pure and Appl. Math., vol. 230, Marcel Dekker Inc., New York, 2002, pp. 173–190. [24] J. M. M. Mart´ınez-Montejano, Zero-dimensional closed set aposyndesis and hyperspaces, Houston J. Math. 32 (2006), no. 4, 1101–1105. [25] S. Mac´ıas, Fans whose hyperspaces are cones, Topology Proc. 27 (2003), no. 1, 217–222. [26] S. Mac´ıas and S. B. Nadler, Jr., n-fold hyperspaces, cones and products, Topology Proc. 26 (2001/02), no. 1, 255–270. [27] V. Mart´ınez-de-la-Vega, An uncountable family of metric compactifications of the ray with remainder pseudo-arc, Topology Appl. 135 (2004), 207–213. [28] J. van Mill and G. M. Reed (eds.), Open problems in topology, North-Holland, Amsterdam, 1990. [29] S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker Inc., New York, 1978. [30] S. B. Nadler, Jr., Continuum theory, Marcel Dekker Inc., New York, 1992. [31] S. B. Nadler, Jr., The fixed point property for continua, Textos, vol. 30, Sociedad Matem´ atica Mexicana, 2005. [32] S. B. Nadler, Jr. and J. T. Rogers, Jr., A note on hyperspaces and the fixed point property, Colloq. Math. 25 (1972), 255–257. [33] J. Oledzki, On symmetric products, Fund. Math. 131 (1988), no. 3, 185–190. [34] F. Orozco-Zitli, Sequential decreasing Whitney properties, Continuum theory (Denton, TX, 1999), Lecture Notes in Pure and Appl. Math., vol. 230, Marcel Dekker Inc., New York, 2002, pp. 297–306. [35] P. Pellicer-Covarrubias, The hyperspaces C(p, X), Topology Proc. 27 (2003), no. 1, 259–285. [36] P. Pellicer-Covarrubias, Retractions in hyperspaces, Topology Appl. 135 (2004), 277–291. [37] H. Whitney, Regular families of curves, I, Proc. Natl. Acad. Sci. 18 (1932), 275–278.


Inverse limits and dynamical systems W. T. Ingram 1. Introduction Throughout this article, we use the term continuum to mean a compact, connected subset of a metric space; by a mapping we mean a continuous function. A continuum is decomposable provided it is the union of two of its proper subcontinua; a continuum is indecomposble if it is not decomposable. A continuum is hereditarily decomposable if each of its subcontinua is decomposable. If X1 , X2 , X3 , . . . is a sequence of metric spaces and f1 , f2 , f3 , . . . is a sequence of mappings such that fi : Xi+1 → Xi for i = 1, 2, 3, . . ., by the inverse limit Q of the inverse limit sequence {Xi , fi } is meant the subset of the product space i>0 Xi that contains the point (x1 , x2 , x3 , . . .) if and only if fi (xi+1 ) = xi for each positive integer i. The inverse limit of the inverse limit sequence {Xi , fi } is denoted by lim{Xi , fi }. For convenience of notation, we will use boldface characters to denote ←− sequences. Thus, if s1 , s2 , s3 , . . . is a sequence, we denote this sequence by s. By this convention, the point (x1 , x2 , x3 , . . .) of an inverse limit space will also be denoted by x, the sequence of factor spaces by X and the sequence of bonding maps by f . For brevity, we will denote the inverse limit space by lim f . ←− A problem set is invariably personal and reflects the interests of the compiler of the set. So it is with this collection of problems. Because of recent developments in the use of inverse limits in certain kinds of models in economics, in Section 7 we include some problems arising from this although we have not personally contributed anything to these applications. Instead we rely on some who have made contributions for problems that reflect the current state of this research. 2. Characterization of chainability Although it is not the original definition of chainability we take as our definition that a continuum is chainable to be that the continuum is homeomorphic to an inverse limit on intervals; a continuum is tree-like provided it is homeomorphic to an inverse limit on trees. A continuum is unicoherent provided it is true that if it is the union of two subcontinua H and K then H ∩ K is connected; a continuum is hereditarily unicoherent provided every subcontinuum of it is unicoherent. A continuum M is a triod provided there is a subcontinuum H of M so that M \ H has at least three components; a continuum is atriodic provided it contains no triod. It is immediate that chainable continua are tree-like. It is well known that chainable continua are atriodic and tree-like continua are hereditarily unicoherent. Several characterizations of chainability of a continuum exist. These include (1) (the original definition) for each ε > 0 there is a finite collection of open sets C1 , C2 , . . . , Cn covering M such that diam(Ci ) < ε for 1 ≤ i ≤ n and Ci ∩ Cj 6= ∅ if and only if |i − j| ≤ 1 and (2) for each positive number ε there is a map fε of the continuum to [0, 1] such that if t is in [0, 1] then the diameter of fε−1 (fε (t)) is less 289

290

§33. Ingram, Inverse limits and dynamical systems

than ε. Notably missing is a characterization involving a list of internal topological properties of the continuum. For example, in case the continuum is hereditarily decomposable, RH Bing [3, Theorem 11] proved that the continuum is chainable if and only if it is atriodic and hereditarily unicoherent. This characterization for hereditarily decomposable continua is satisfying in that it is given in terms of internal topological properties of the continuum. 638?

Problem 1. Characterize chainability of a continuum in terms of internal topological properties of the continuum. J.B. Fugate [10] extended Bing’s result from the class of hereditarily decomposable continua to the class of those continua having the property that every indecomposable subcontinuum is chainable. Thus, Problem 1 may be solved by characterizing chainability of indecomposable continua. Case and Chamberlin [6] gave a characterization of tree-like continua as those one-dimensional continua for which every mapping to a one-dimensional polyhedron is inessential (i.e., homotopic to a constant map). J. Krasinkiewicz later proved that a one-dimensional continuum is tree-like if and only if every mapping of it to a figure-8 (the union of two circles with a one-point intersection) is inessential [26]. Although these characterizations of tree-likeness do not involve “internal” topological properties, it would still be of significant interest to characterize chainability among tree-like continua. Since tree-like continua are hereditarily unicoherent, atriodicity is a natural candidate for one of the properties on a list of characterizing properties. That atriodicity alone is not sufficient was shown in [14]. One significant attempt at characterizing chainability involves the notion of the span of a continuum. If M is a continuum, the span of M is the least upper bound of {ε ≥ 0 : there is a subcontinuum C of M × M such that p1 (C) = p2 (C) and dist(x, y) ≥ ε for all (x, y) in C} (p1 and p2 denote the projections of M × M into its factors). The following problem on span remains open even though it was featured [8] in the first volume of Open problems in topology.

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Problem 2. If the span of a continuum is 0, is M chainable? A. Lelek intoduced span in [28] and proved that chainable continua have span 0. Although span 0 is a topological property, in some real sense it is not “internal”. Consequently, if one were to settle Problem 2 in the affirmative, the nature of the definition of span would, in this author’s opinion, leave work to be done on Problem 1. That said, Problem 2 is significant in its own right and not only because it has become an “old” problem. For instance, a positive solution would tell us that we know all of the homogeneous plane continua [34]. 3. Plane embedding In thinking about Problem 1 and in light of the Case–Chamberlin theorem [6] characterizing tree-likeness, the author began a quest to settle the question whether atriodic tree-like continua are chainable. That investigation led to an example of an atriodic tree-like continuum that is not chainable [14]. Span turned out to be just the tool needed to show that the example obtained is not chainable.

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B B

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Figure 1. Bing’s example of a nonplanar tree-like continuum However, span was not the first tool the author tried to use. In fact, two other properties of chainable continua first came to mind: planarity and the fixed point property. Bing showed that chainable continua can be embedded in the plane [3] and O.H. Hamilton showed that chainable continua have the fixed point property [12]. The author chose to try to employ Bing’s result and construct an atriodic tree-like continuum that cannot be embedded in the plane. This leads to our next problem. Problem 3. Characterize those tree-like continua that can be embedded in the plane. The reader interested in this problem should be familiar with Brian Raines’ work [35] on local planarity of inverse limits of graphs. Of course, tree-like continua that cannot be embedded in the plane are well known. Arguably the simplest of these may be one given by Bing [3]. This example consists of a ray with remainder a simple triod together with an arc that intersects the union of the ray and the triod only at the junction point of the triod. A map of the 4-od that produces in its inverse limit a simple triod and a ray having the simple triod as a remainder is shown in Figure 1. Bing’s example is obtained by attaching an arc to this inverse limit at the point (O, O, O, . . .) and otherwise misses the inverse limit. Finding an atriodic example presents somewhat more of a challenge. Although other non-planar atriodic tree-like continua are known, an example may be constructed in the following way. Let M be the atriodic tree-like continuum with positive symmetric span that the author constructed in [14] and let C be the product of M with a Cantor set. A construction of Laidacker [27] produces an atriodic tree-like continuum M 0 that contains C. Duˇsan Repovˇs, Arkadij B. Skopenkov, ˇcepin prove in [36] that the plane does not contain uncountably and Evgenij V. Sˇ

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A O

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Figure 2. A first attempt at a nonplanar atriodic tree-like continuum

many mutually exclusive tree-like continua with positive symmetric span so the continuum M 0 is non-planar. In order to tackle Problem 3, it would be helpful to have some examples of planar continua that “look” like they might be non-planar as well as some simpler examples of non-planar atriodic tree-like continua to study. The remainder of this section is devoted to some examples and possible examples. The author’s first attempts to construct an atriodic tree-like continuum that cannot be embedded in the plane failed (as have many subsequent attempts). We briefly describe an early attempt. The picture in Figure 2 is a schematic drawing of a mapping f of a simple triod T = [OA] ∪ [OB] ∪ [OC] onto itself. The action of the function is to take the first half of [OA] onto [AO] with f (O) = A and the second half of [OA] onto [OB] with f (A) = B; f takes the first third of [OB] onto [AO], the next sixth onto the first half of [OC], the next sixth folds back to O and the final third is taken onto [OB]; f takes the first third of [OC] onto [AO], the next third half way out [OB] and back, and the final third onto [OC]. The resulting inverse limit is atriodic, but it is a chainable continuum because f ◦ f factors through [0, 1] (i.e., there are maps g : T → [0, 1] and h : [0, 1] → T so that f = h ◦ g). Though the schematic of f cannot be drawn in the plane, lim f being ←− chainable is planar. An alternative to Bing’s non-planar tree-like continuum mentioned above is the following. Take a continuum consisting of two mutually exclusive rays each having the same simple triod as remainder but the rays wind around the triod in opposite directions. The resulting tree-like continuum is non-planar. This observation suggests the following way possibly to construct a non-planar atriodic tree-like continuum. The continuum is an inverse limit on a simple 5-od, [OA] ∪ [OB] ∪ [OC] ∪ [OD] ∪ [OE]. Restricted to the triod [OA] ∪ [OB] ∪ [OC] our 5od map is just the triod map that the author used in [14] to obtain an atriodic tree-like continuum that is not chainable. We use the other two arms of the 5-od to obtain rays that wind in “opposite” directions onto that example. We provide a schematic diagram of the map in Figure 3. The author does not know if the resulting inverse limit is non-planar.

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Figure 3. A nonplanar example?

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Figure 4. A twisted example

There is a somewhat simpler possibility that results from an inverse limit on 4-ods. The author does not know if the resulting inverse limit space is nonplanar. The bonding map f (shown in a schematic in Figure 4) has the interesting feature that, although it can be “drawn in the plane”, f 2 cannot be “drawn in the plane”. This appears to be caused by a twist of the arms of the 4-od imposed by the bonding map. Unfortunately, as our second example shows (see Figure 2), not being able to “draw” a schematic of the bonding map in the plane does not guarantee that the inverse limit is non-planar.

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4. Inverse limits on [0, 1] Considerable interaction between dynamacists and continuum theorists has occurred in the past fifteen or twenty years. Inverse limits appeal to dynamacists in part because they allow one to transform the study of a dynamical system consisting of a space and a mapping of that space into itself into the study of a (perhaps more complicated) space, the inverse limit, and a homeomorphism, the shift, of that space into itself. Considerations in dynamics have led to extensive investigations of parameterized families of maps. Many of these are maps of [0, 1] into itself and include the logistic family and the tent family. Interest in these families also rekindled the author’s interest in inverse limits on [0, 1] using a constant sequence of bonding maps in which that bonding map is chosen from one of those two families or from one of several other families of piecewise linear maps including the families ft for 0 ≤ t ≤ 1, gt for 0 ≤ t ≤ 1, fab (also denoted gbc by the author and others) where both parameters come from [0, 1], and the family of permutation maps. In this article we provide definitions only for the tent family (in the next paragraph) and the permutation maps (in the next section). For definitions of the families not discussed further in this article and information on some of the inverse limits generated by these families see [15–17, 21]. With one exception these are families of unimodal maps, a class of maps of special interest in dynamics. Permutation maps are Markov maps, a class also of interest in dynamics. A map is monotone provided its point inverses are connected; a map f : [0, 1] → [0, 1] is unimodal provided f is not monotone and there is a point c, 0 < c < 1, such that f [0, c] and f [c, 1] are monotone. A map f : [0, 1] → [0, 1] is Markov provided there is a finite subset {x1 = 0, x2 , . . . , xn = 1} with xi < xi+1 and f [xi , xi+1 ] is monotone for 1 ≤ i < n. Tent maps are unimodal maps of [0, 1] constructed as follows. Choose a number s from [0, 1] and let fs : [0, 1] → [0, 1] be the piecewise linear map that passes through (0, 0), (1/2, s), and (1, 0). (Specifically, fs is given by fs (x) = 2sx for 0 ≤ x ≤ 1/2 and fs (x) = (2 − 2x)s for 1/2 ≤ x ≤ 1.) One problem involving the tent family has sparked considerable interest and has given rise to a large number of partial results. 641?

Problem 4. If fs and ft are tent maps with lim fs and lim ft homeomorphic, is ←− ←− s = t? This has been settled in a number of cases including Lois Kailhofer’s proof ˇ for maps that have periodic critical points [24]. Stimac has announced a positive solution if the maps have preperiodic critical points. The collection of inverse limits arising from the tent family is rich in its variety. Barge, Brucks, and Diamond have shown that there are uncountably many parameter values at which the inverse limit is so complicated that it contains a copy of every continuum arising as an inverse limit space from a tent family core (see the next paragraph) [2]. In spite of the presence of complicated topology at some parameter values, progress has been made on Problem 4 when the orbit of the critical point is infinite. B. Raines has begun a systematic study of these inverse

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limits and has made some significant progress for certain parameter values. The author acknowledges private correspondence with Professor Raines that provided some of the problems in this section as well as some of the information on the literature related to these problems. If fs is a tent map lim fs is the closure of a topological ray. Except for s = 1 ←− the inverse limit is a decomposable continuum and if R is the ray that is dense in the inverse limit, R − R is a proper subcontinuum that results from the inverse limit on [fs (s), s] using the restriction of fs to that interval as the bonding map. We refer to lim fs [fs (s), s] as the core of lim fs and the map fs [fs (s), s] as a tent ←− ←− core. Sometimes the tent core is rescaled to be the map of [0, 1] onto itself given by fs (x) = sx + 2 − s for 0 ≤ x ≤ 1 − 1/s and fs (x) = s − sx for 1 − 1/s ≤ x ≤ 1. Since the critical point is different depending on one’s perspective, it is simply denoted by c in the remainder of this section. Raines’ approach to the case that the orbit of the critical point c is infinite has T∞ been to look at the ω-limit set of c, ω(c) = n=1 {f k (c) : k ≥ n}. When the orbit of c is infinite, ω(c) = [0, 1] or ω(c) is totally disconnected. If the orbit is infinite and ω(c) is totally disconnected, ω(c) may be a countable set, a Cantor set, or the union of a countable set and a Cantor set. It is in the case that ω(c) = [0, 1] that the Barge, Brucks and Diamond phenomenon of [2] occurs (i.e., there are parameter values at which the inverse limit of the tent map contains a copy of every continuum that arises as an inverse limit space from a tent family core). Problem 5 (Raines). Suppose f is a tent core with critical point c such that ω(c) = [0, 1]. If C is a composant of lim f , does C contain a copy of every continuum that ←− arises as an inverse limit space of a tent family core?

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Problem 6 (Raines). Suppose f is a unimodal map with critical point c. Give necessary and sufficient conditions on c so that lim f contains a copy of every ←− continuum that arises as an inverse limit space in a tent family core.

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In case f is a tent core with critical point c and ω(c) is countable or the union of a countable set and a Cantor set, it is known that the inverse limit is an indecomposable arc continuum without end points (by an arc continuum we mean a continuum such that every proper subcontinuum is an arc). Good, Knight, and Raines have shown [11] that there are uncountably many members of the tent family cores with ω(c) countable that have topologically different inverse limits. In case f is a tent core with critical point c and ω(c) is a Cantor set, the inverse limit is indecomposable but it may have end points. If it has end points the set of end points is uncountable [5]. The subcontinua of lim f can be quite ←− complicated as demonstrated in [4]. This gives rise to the next problem. Problem 7 (Raines). Let f be a tent core with critical point c and ω(c) a Cantor set. Classify all possible subcontinua of lim f . ←− We close this section with one final problem. If n is a positive integer and σ is a permutation on the set {1, 2, . . . , n}, define a map fσ : [0, 1] → [0, 1] in the following way: (1) for 1 ≤ i ≤ n, let ai = (i − 1)/(n − 1), (2) let fσ (ai ) = aσ(i) , and

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(3) extend fσ linearly to all of [0, 1]. We call a map so constructed a permutation map. These maps are all Markov maps and many interesting continua result as the inverse limit space based on a permutation map. In [18] the author began a study of the inverse limits spaces that result from using a permutation map in an inverse limit. By brute force, all continua arising from permutations based on 3, 4, or 5 elements were determined. 645?

Problem 8. Classify all continua arising from permutation maps. 5. The property of Kelley A continuum M with metric d is said to have the Property of Kelley provided if ε > 0 there is a positive number δ such that if p and q are points of M and d(p, q) < δ and H is a subcontinuum of M containing p then there is a subcontinuum K of M containing q such that H(H, K) < ε (H denotes the Hausdorff distance on the hyperspace of subcontinua C(M )). This property that we now call the Property of Kelley was introduced by J. Kelley in his study of hyperspaces, but it is a nice continuum approximation property in its own right. The author considered the property in [19, 20, 23]. While presenting the results that appeared in [19, 20] in seminar, the author was asked the following question by W.J. Charatonik.

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Problem 9 (Charatonik). Is there a characterization of the Property of Kelley in terms of the inverse limit representation of the continuum? The author briefly tried to distill a sufficient condition from the proofs in the papers in [19, 20] but never found a satisfying theorem. Nonetheless, it would be of interest to be able to determine the presence of the Property of Kelley based on some easily checked conditions on the bonding maps in an inverse limit representation of the continuum. Private communication with W.J. Charatonik indicates that he and a student have obtained some sufficient conditions on an inverse limit sequence to guarantee that the inverse limit have the Property of Kelley. Permutation maps were defined in Section 4. In [18] it was shown that if f is a permutation map based on a permutation on 3, 4, or 5 elements, then lim f has ←− the Property of Kelley. This leads us to ask the following question.

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Problem 10. Do all permutation maps produce continua with the Property of Kelley? 6. Inverse limits with upper semi-continuous bonding functions W.S. Mahavier introduced inverse limits with upper semi-continuous bonding functions in [29] but as inverse limits on closed subsets of [0, 1] × [0, 1]. In that article he showed that inverse limits on closed subsets of [0, 1] × [0, 1] are inverse limits on [0, 1] using upper semi-continuous closed set valued functions as bonding functions. In a subsequent paper [22], Mahavier and the author extended the definition to the setting of inverse limits on compact Hausdorff spaces using upper semi-continuous closed set valued bonding functions. If Y is a compact Hausdorff

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space, 2Y denotes the collection of all closed subsets of Y . If X and Y are compact Hausdorff spaces, a function f : X → 2Y is called upper semi-continuous at the point x of X provided if O is an open set in Y that contains f (x) then there is an open set U in X that contains x and f (t) is a subset of O for every t in U . If X1 , X2 , X3 , . . . is a sequence of compact Hausdorff spaces and f1 , f2 , f3 , . . . is a sequence of upper semi-continuous functions such that fi : Xi+1 → 2Xi for Qeach i, by the inverse limit of the inverse sequence {Xi , fi } is meant the subset of i>0 Xi that contains the point x = (x1 , x2 , x3 , . . .) if and only if xi ∈ f (xi+1 ). The reader will note that in case the functions are single valued, this definition reduces to the usual definition of an inverse limit. Beyond the collection of chainable continua that occur with single valued bonding functions, many interesting examples of continua result from inverse limits on [0, 1] with upper semi-continuous bonding functions that cannot occur with single valued functions. Among these are the Hilbert cube, the Cantor fan, a 2-cell with a sticker, and the Hurewicz continuum H that has the property that if M is a metric continuum there is a subcontinuum K of H and a mapping of K onto M . The example that produces a 2-cell with an attached arc leads to the following problem. Problem 11. Is there an upper semi-continuous function f : [0, 1] → 2[0,1] such that lim f is a 2-cell? ←− Admittedly, this problem is rather more specific than most in this article, but perhaps it can serve as a starting point for an interesting investigation of these new and different inverse limits. We end this section with a problem inspired by considerations from Section 7. Some models in economics are not well-defined either forward in time or backward in time [7], [37]. Some models consist of the union of two mappings that have no point in common. Perhaps an investigation of these new inverse limits using these models would be helpful to economists as well as a way to begin work on our next problem. Problem 12. Suppose f : [0, 1] → 2[0,1] is an upper semi-continuous function that is the union of two mappings of [0, 1]. What can be said about lim f ? ←− 7. Applications of inverse limits in Economics An exciting recent development in inverse limits is the development of models in economics in which the state of the model at time t is related to its state at time t + 1 by some non-invertible mapping f . A solution to the model is an infinite sequence x1 , x2 , x3 , . . . such that f (xt+1 ) = xt for t = 1, 2, 3, . . .. So the set of solutions is the inverse limit on the state space using the map f as a bonding map. These models have arisen in cash-in-advance models [25] and overlapping generations models [30, 31] studied by various economists. These models generally fall into a category of models described by economists as having “backward dynamics” or as models involving “backward maps”. Economists are interested in the inverse limit because it contains as its points all future states predicted by the model. The author acknowledges private correspondence with

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Judy Kennedy and Brian Raines used in the development of this section and appreciates the contribution of problems by both of them. The problems that they contributed are labeled below with their names. Of course, any errors or misstatement of problems are solely the responsibility of the author. If f : X → X and g : X → X are maps of a topological space X, we say that f and g are topologically conjugate provided there is a homeomorphism h : X → X such that f ◦ h = h ◦ g. If f and g are topologically conjugate, a homeomorphism h such that f ◦ h = h ◦ g is called a conjugacy. 650?

Problem 13 (Kennedy–Stockman). Suppose f : [0, 1] → [0, 1] and g : [0, 1] → [0, 1] are topologically conjugate. How does one construct a homeomorphism h so that f ◦ h = h ◦ g? This problem deserves attention independent of the interest by economists. For economists the existence of a conjugacy is not sufficient information for carrying out some of the computations they need such as the computation of measures and then integrals for utility functions. Specific questions related to this problem and asked by Kennedy and Stockman include: (1) Can the conjugacy be constructed by means of a sequence of approximations? (2) If f and g are piecewise differentiable, must the conjugacy be piecewise differentiable? The next problem is related to Problem 5 above.

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Problem 14 (Kennedy–Stockman). Do continua that contain copies of every inverse limit that arises in a tent family core occur as inverse limits in the cashin-advance model [25] or the overlapping generations model [33]? Some models in economics are based on relations instead of functions so neither forward nor backward dynamics is well defined. In particular the Christiano– Harrison model [7] and a Stockman model [37] fit this scenario. Perhaps inverse limits with upper semi-continuous bonding functions (see Section 6) could be employed in an analysis of these models. Consequently, we reiterate Problem 12. In considering models in economics, measure theory will likely play an important role for several reasons one of which we have already mentioned. For instance, when economists consider models involving backward dynamics, they would like to be able to “rank” the inverse limit spaces in some meaningful way, perhaps by using “natural” invariant measures. For a survey of literature on such measures see [13]. When comparing two inverse limit spaces but with a precise meaning of “better” to be determined, Kennedy and Stockman ask the following.

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Problem 15 (Kennedy–Stockman). Suppose policy A and policy B in economics lead to different inverse limit spaces. Determine which of the inverse limits is “better”. With a precise meaning of “complex” to be determined, they also ask.

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Problem 16 (Kennedy–Stockman). For an economics model, what is the measure of the set of initial conditions that lead to “complex” dynamics?

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Problem 17 (Kennedy–Stockman). In a model from economics, if an equilibrium point (i.e., point in the inverse limit) is chosen at random, what is the probability that it is “complex”?

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One search for appropriate measures on the inverse limit space centers on somehow making use of measures already developed. Kennedy and Stockman have recently succeeded in “lifting” given measures for interval maps to measures on the corresponding inverse limit spaces although they remark that such measures on the inverse limit space apparently are already known, see [9]. For an introduction to measures for interval maps see [1, Sections 6.4–6.6]. See also [13]. Kennedy and Stockman ask if there exist other useful measures one might consider, particularly in non-chaotic situations. Recall that if f : X → X is a mapping of a metric space and A is a closed subset of X such that f [A] = A, then we call A an invariant set. If A is a closed invariant subset of X, then the basin of attraction of A is {x ∈ X : ω(x) ⊂ A}. A subset B of X is nowhere dense in X provided B does not contain a nonempty open set. A subset M of X is said to be residual in X provided X −M is the union of countably many nowhere dense subsets. A closed invariant subset of X is called a topological attractor [33] for f provided the basin of attraction for A contains a residual subset of X and if A0 is another closed invariant subset of X then the common part of the basin of attraction of A0 and the basin of attraction of A is the union of at most countably many nowhere dense sets. For more information of topological attractors and metric attractors (defined below), see [33]. One possible tool for analyzing an inverse limit arising in a model from economics lies in the shift homeomorphism. There are two shifts and they are inverses of each other. Specifically, below we are referring to the shift σ : lim f → lim f ←− ←− given by σ(x) = (x2 , x3 , x4 , . . .). Raines asks the following. Problem 18 (Raines). Let f be a map of the interval. Find necessary and sufficient conditions for lim f to admit a proper subset that is a topological attractor ←− for the shift homeomorphism.

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Problem 19 (Raines). Let f be a unimodal map of the interval. Classify all of the topological attractors for the shift homeomorphism on lim f . ←− Not all models from economics involve one-dimensional spaces. This prompts the following problem.

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Problem 20 (Raines). Let f be a map of [0, 1] × [0, 1]. Identify topological attractors in lim f under the shift homeormorphism. ←− If X is a metric space with a measure µ, f : X → X is a mapping and A is a closed invariant subset of X, then A is called a metric attractor for f provided the basin of attraction for A has positive measure and and if A0 is another closed invariant subset of X then the common part of the basin of attraction of A0 and the basin of attraction of A has measure zero.

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Problem 21 (Raines). In the previous two problems, change the phrase “topological attractor” to “metric attractor”. References [1] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos, Springer, New York, 1997. [2] M. Barge, K. Brucks, and B. Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3563–3570. [3] R H Bing, Snake-like continua, Duke Math. J. 18 (1951), 653–663. [4] K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math. 160 (1999), no. 3, 219–246. [5] H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl. 96 (1999), no. 3, 191–208. [6] J. H. Case and R. E. Chamberlin, Characterizations of tree-like continua, Pacific J. Math. 10 (1960), 73–84. [7] L. Christiano and S. Harrison, Chaos, sunspots and automatic stabilizers, J. Monetary Economics 44 (1999), 3–31. [8] H. Cook, W. T. Ingram, and A. Lelek, Eleven annotated problems about continua, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 295–302. [9] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic theory, Springer, New York, 1982. [10] J. B. Fugate, Decomposable chainable continua, Trans. Amer. Math. Soc. 123 (1966), 460– 468. [11] C. Good, R. Knight, and B. Raines, Non-hyperbolic one-dimensional invariant sets with uncountably infinite collections of inhomogeneities, Preprint. [12] O. H. Hamilton, A fixed point theorem for pseudo-arcs and certain other metric continua, Proc. Amer. Math. Soc. 2 (1951), 173–174. [13] B. R. Hunt, J. A. Kennedy, T.-Y. Li, and H. E. Nusse, SLYRB measures: natural invariant measures for chaotic systems, Phys. D 170 (2002), no. 1, 50–71. [14] W. T. Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972), no. 2, 99–107. [15] W. T. Ingram, Inverse limits, Sociedad Matem´ atica Mexicana, M´ exico, 2000. [16] W. T. Ingram, Inverse limits on [0, 1] using piecewise linear unimodal bonding maps, Proc. Amer. Math. Soc. 128 (2000), no. 1, 279–286. [17] W. T. Ingram, Inverse limits on circles using weakly confluent bonding maps, Topology Proc. 25 (2000), Spring, 201–211. [18] W. T. Ingram, Invariant sets and inverse limits, Topology Appl. 126 (2002), no. 3, 393–408. [19] W. T. Ingram, Inverse limits and a property of J. L. Kelley. I, Bol. Soc. Mat. Mexicana (3) 8 (2002), no. 1, 83–91. [20] W. T. Ingram, Inverse limits and a property of J. L. Kelley. II, Bol. Soc. Mat. Mexicana (3) 9 (2003), no. 1, 135–150. [21] W. T. Ingram and W. S. Mahavier, Interesting dynamics and inverse limits in a family of one-dimensional maps, Amer. Math. Monthly 111 (2004), no. 3, 198–215. [22] W. T. Ingram and W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), no. 1, 119–130. [23] W. T. Ingram and D. D. Sherling, Two continua having a property of J. L. Kelley, Canad. Math. Bull. 34 (1991), no. 3, 351–356. [24] L. Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math. 177 (2003), no. 2, 95–120. [25] J. Kennedy, D. R. Stockman, and J. A. Yorke, Inverse limits and an implicitly defined difference equation from economics, Topology Appl. , To appear. [26] J. Krasinkiewicz, On one-point union of two circles, Houston J. Math. 2 (1976), no. 1, 91–95. [27] M. Laidacker, Imbedding compacta into continua, Topology Proc. 1 (1977), 91–105.

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[28] A. Lelek, Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199–214. [29] W. S. Mahavier, Inverse limits with subsets of [0, 1] × [0, 1], Topology Appl. 141 (2004), 225–231. [30] A. Medio, The problem of backward dynamics in economics models, 2000, Preprint. [31] A. Medio, Invariant probability distributions in economic models: a general result, Macroeconomic Dynamics 8 (2004), 162–187. [32] A. Medio and B. Raines, Backward dynamics in economics. The inverse limit approach, J. Economic Dynamics and Control (2006), Article in press. [33] A. Medio and B. Raines, Inverse limit spaces arising from problems in economics, Topology Appl. (2006), Article in press. [34] L. G. Oversteegen and E. D. Tymchatyn, Plane strips and the span of continua. I, Houston J. Math. 8 (1982), no. 1, 129–142. [35] B. Raines, Local planarity in one-dimensional continua, Preprint. ˇcepin, On uncountable collections of continua and [36] D. Repovˇs, A. B. Skopenkov, and E. V. Sˇ their span, Colloq. Math. 69 (1995), no. 2, 289–296. [37] D. R. Stockman, Balanced-budget rules: cycles and complex dynamics, Preprint.


Indecomposable continua Wayne Lewis Except for discussion near the end of this article, we shall consider a continuum to be a compact connected metric space. In discussion near the end of this article we shall consider a non-metric continuum to be a compact connected nonmetrizable Hausdorff space. A continuum is indecomposable if it is not the union of two proper subcontinua. It is hereditarily indecomposable if each of its subcontinua is indecomposable. Hereditary equivalence The best studied hereditarily indecomposable continuum is the pseudo-arc. Moise [62] showed that the pseudo-arc is hereditarily equivalent, i.e., homeomorphic to each of its nondegenerate subcontinua, and gave it its name because it shares this property with the arc. Question 1. Is the pseudo-arc the only nondegenerate continuum other than the arc which is hereditarily equivalent?

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It follows from results by Henderson [26] and Cook [18] that any nondegenerate hereditarily equivalent continuum other than the arc must be hereditarily indecomposable and tree-like. Mohler and Oversteegen [61] have constructed examples of non-metric decomposable hereditarily equivalent continua, including one which is not a Hausdorff arc. Smith [89] has constructed a non-metric hereditarily indecomposable hereditarily equivalent continuum, obtained as an inverse limit of ω1 copies of the pseudo-arc. Oversteegen and Tymchatyn [65] have shown that any planar hereditarily equivalent continuum must be close to being chainable, i.e., must be weakly chainable and have symmetric span zero. Homogeneity Bing [9] has characterized the pseudo-arc as a nondegenerate hereditarily indecomposable chainable continuum, i.e., any such continuum must be homeomorphic to the continuum constructed by Moise, and to a continuum constructed earlier by Knaster [32] to show that hereditarily indecomposable continua exist. Bing [7] and Moise [63] have independently shown that the pseudo-arc is homogeneous and Bing [10] has shown that it is the only nondegenerate homogeneous chainable continuum. This latter result has been generalized by this author [43] to show that the pseudo-arc is the only nondegenerate homogeneous almost chainable ontinuum. The first characterization of the pseudo-arc has been used extensively and it would be useful to know if it can be generalized. Question 2. Is the pseudo-arc the only nondegenerate hereditarily indecomposable weakly chainable continuum? 303

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A weakly chainable continuum can be described in terms of a defining sequence of open covers. Fearnley [22] and Lelek [38] have shown that a continuum is weakly chainable if and only if it is the continuous image of a chainable continuum, and hence of the pseudo-arc. Thus the above question can be rephrased as “If X is a nondegenerate hereditarily indecomposable continuum which is the continuous image of the pseudo-arc, is X itself a pseudo-arc?” This question is of interest not just in terms of the images of the pseudo-arc or a possible generalization of a known characterization of the pseudo-arc. It is central to the classification of homogeneous plane continua and is a special case of a family of questions of interest. 662?

Question 3. Is the pseudo-arc the only nondegenerate homogeneous nonseparating plane continuum? Hagopian [23] has shown that every nonseparating homogeneous plane continuum is hereditarily indecomposable. Oversteegen and Tymchatyn [66] have shown that every such continuum is weakly chainable. More on the status of the classification of homogeneous plane continua or homogeneous one-dimensional continua can be found in survey articles by this author [51] and by Rogers [79, 80]. Rogers [77] has shown that every homogeneous hereditarily indecomposable continuum is tree-like. Krupski and Prajs [36] have shown that every homogeneous tree-like continuum is hereditarily indecomposable. Both of these results are independent of whether the continuum is planar.

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Question 4. Is the pseudo-arc the only nondegenerate homogeneous tree-like continuum? While it is known from the result of Oversteegen and Tymchatyn that any such continuum which is planar must be weakly chainable, such is not yet known to be the case for the possibly nonplanar case.

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Question 5. Is every homogeneous tree-like continuum weakly chainable? A continuous surjection f : X → Y between continua is confluent if, for each subcontinuum H of Y and each component C of f −1 (H), f (C) = H, The class of confluent maps includes the classes of open maps and of monotone maps. Cook [16] has shown that a continuum Y is hereditarily indecomposable if and only if every continuous surjection from a continuum onto Y is confluent. Thus, any hereditarily indecomposable continuum which is weakly chainable is the confluent image of the pseudo-arc. McLean [57] has shown that the confluent image of a tree-like continuum is tree-like. A positive answer to the following question would show that the pseudo-arc is the only nondegenerate hereditarily indecomposable weakly chainable continuum, providing positive answers to Questions 2 and 3.

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Question 6. Is the confluent image of a chainable continuum chainable? Bing [9] and Rosenholtz [81], respectively, have shown that monotone maps and open maps preserve chainability. Confluent maps also preserve indecomposability, hereditary indecomposability and atriodicity. One of the most general forms of this family of questions is due to Mohler [60].

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Question 7. Is every weakly chainable atriodic tree-like continuum chainable?

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Minc [59] has provided a partial answer by showing that any atriodic weakly chainable continuum which is an inverse limit of trees with simplicial bonding maps is chainable. As indicated above, Rogers [77] has shown that every homogeneous hereditarily indecomposable continuum is tree-like. However, there are homogeneous indecomposable continua which are not tree-like, e.g., solenoids or solenoids of pseudoarcs. However, all such known nondegenerate continua are one-dimensional, leading Rogers to ask the following two questions. (Any product of nondegenerate continua is aposyndetic and hence decomposable. Thus there can be no indecomposable homogeneous analog of the Hilbert cube.) Question 8 ([79]). Is each nondegenerate homogeneous indecomposable continuum one-dimensional?

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Question 9 ([78]). Is each homogeneous indecomposable cell-like continuum treelike?

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Every known nondegenerate homogeneous indecomposable continuum (whether hereditarily indecomposable or not) is circle-like. Also, every known homogeneous plane continuum (whether indecomposable or not, whether separating the plane or not) is circle-like. The family of homogeneous circle-like continua has been completely classified [25, 46, 74] as the circle, solenoids, circle of pseudo-arcs and solenoids of pseudo-arcs, with the circle of pseudo-arcs and for each solenoid the solenoid of pseudo-arcs being unique. Question 10. Is every nondegenerate homogeneous indecomposable continuum circle-like?

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Question 11. Is every nondegenerate homogeneous plane continuum circle-like?

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Burgess [14] has shown that every continuum which is both circle-like and tree-like is chainable. Recalling Bing’s result [10] that the only nondegenerate homogeneous chainable continuum is the pseudo-arc, positive answers to each of Question 9 and 10 would imply that the only nondegenerate indecomposable homogeneous cell-like continuum is the pseudo-arc while a positive answer to Question 10 implies a positive answer to Question 11. A positive answer to Question 11 implies that the classification of homogeneous plane continua is complete with the four known examples of the point, simple closed curve, pseudo-arc and circle of pseudo-arcs. A space X is said to be homogeneous with respect to the class M of maps if for every x, y ∈ X there exists a continuous surjection f : X X of X onto itself with f (x) = y and f ∈ M. (Usual homogeneity is thus homogeneity with respect to homeomorphisms.) J. Charatonik and Maćkowiak [15] have shown that the result of Bing can be strengthened to show that the pseudo-arc is the only nondegenerate chainable continuum which is homogeneous with respect to confluent maps. Several examples are known of continua which are not homogeneous but which


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are homogeneous with respect to various classes of maps. Prajs [71] has shown that the disc is homogeneous with respect to open maps, while Seaquist [82] has shown that it is not homogeneous with respect to monotone maps. Prajs [72] and Seaquist [83] have independently shown that the Sierpi´ nski universal plane curve is homogeneous with respect to open monotone maps. J. Charatonik has asked the following question, with special interest in possible generalized homogeneity of the pseudo-circle. 671?

Question 12. Does there exist an hereditarily indecomposable continuum which is not homogeneous but which is homogeneous with respect to continuous functions (and hence with respect to confluent maps)? -premaps Determining whether a nondegenerate weakly chainable hereditarily indecomposable continuum must be chainable, and hence a pseudo-arc, is equivalent to determining if such a continuum can be mapped onto a chainable continuum with arbitrarily small point inverses. The pseudo-arc has a strong version of a converse property. If f : P → X is a continuous surjection from the pseudo-arc P onto the nondegenerate continuum X and > 0, there exists a homeomorphism h : P → P such that diam(f ◦ h)−1 (x) < for every x ∈ X, i.e., f ◦ h is an -map of P onto X. For such a continuous surjection f we shall call the composition f ◦ h an -premap corresponding to f. It can be shown that any nondegenerate continuum sharing this property with the pseudo-arc P that every continuous surjection onto a nondegenerate continuum has a corresponding -premap must be chainable and indecomposable.

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Question 13. Is the pseudo-arc the only nondegenerate continuum with the property that for every > 0 every continuous surjection onto a nondegenerate continuum has a corresponding -premap? Fixed points Indecomposable continua play an important role in the study of the fixedpoint property. We expect this topic to be more thoroughly covered in a separate article in this volume and include only the following questions. The first two are due to Lysko [54].

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Question 14. Does there exist a continuum X with the fixed point property such that X × P (P = pseudo-arc) does not have the fixed point property? There are known examples [11] of continua with the fixed point property whose product with the unit interval [0, 1] does not have the fixed point property. However, these do not translate to such examples for products with the pseudo-arc and the structure of an hereditarily indecomposable continuum places restrictions on maps of products of it with other continua. The above question is a special case of the following. The case of the pseudo-arc, or a nondegenerate hereditarily

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indecomposable continuum with the fixed point property, seems especially challenging. Question 15. If X is a nondegenerate continuum with the fixed point property, does there always exist a nondegenerate continuum Y with the fixed point property such that X × Y does not have the fixed point property?

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A continuum X has the complete invariance property if every nonempty closed subset of X is the complete fixed point set of some self-map of X. Martin and Nadler [55] have shown that every two-point set is a fixed point set for some continuous self-map of the pseudo-arc. Cornette [20] has shown that every subcontinuum of the pseudo-arc is a retract of the pseudo-arc. Toledo [92] has shown that every subcontinuum of the pseudo-arc is the fixed point set of a periodic homeomorphism of the pseudo-arc. This author [42] has shown that there are proper subsets of the pseudo-arc with nonempty interior which are the fixed point sets of homeomorphisms. The following question due to Martin and Nadler [55] is also of interest in the special case of self-homeomorphisms of the pseudo-arc. Question 16. Does the pseudo-arc have the complete invariance property?

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Maps of products Bellamy and Lysko [6] have shown that every homeomorphism of P ×P , where P is the pseudo-arc, is a composition of a product of homeomorphisms on the individual factors with a permutation of the factors. Bellamy and Kennedy [5] have extended this to a product of an arbitrary number of copies of the pseudo-arc. The arguments in both cases use specific properties of the pseudo-arc. However, the considerations motivating this investigation stem from the structure of hereditarily indecomposable continua in general. Q Question 17. If X = α∈A Xα is a product of hereditarily indecomposable continua, is every homeomorphism of X a composition of a product of homeomorphism on the individual factor spaces with a permutation of the factors?

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The following question about the structure of products of the pseudo-arc is due to Bellamy. Question 18. Does every nondegenerate subcontinuum of P n , the product of finitely many copies of the pseudo-arc, contain a pseudo-arc? A continuum X is pseudo-contractible if there exists a continuum Y , points a and b in Y , a point x0 ∈ X and a continuous function f : X × Y → X such that f (x, a) = x for each x ∈ X and f (x, b) = x0 for each x ∈ X. There exist examples of continua which are pseudo-contractible but not contractible, e.g., the spiral around a disk. Sobolewski [91] has shown that no nondegenerate chainable continuum other than the arc is pseudo-contractible. In particular, the pseudo-arc and the Knaster-type indecomposable continua are not pseudo-contractible.

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Question 19. Does there exist a nondegenerate (hereditarily) indecomposable continuum which is pseudo-contractible? Concerning maps of products of pseudo-arcs, Lysko [53] has also asked the following question.

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Question 20. Assume that P is the pseudo-arc and that r : P ×P → ∆ = {(x, y) ∈ P × P |x = y} is a continuous retraction. Must r be of the form r(x, y) = (x, x) for all (x, y) or r(x, y) = (y, y) for all (x, y)? Homeomorphism groups This author [48] has used properties of homeomorphisms of P ×M, where P is a pseudo-arc and M a continuum, to show that H(P ), the topological group of selfhomeomorphisms of the pseudo-arc, does not contain a nondegenerate continuum. The following is a variation of questions asked by Brechner [12], Krasinkiewicz [35] and this author [45].

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Question 21. Is H(P ), the topological group of all self-homeomorphisms of the pseudo-arc P , totally disconnected?

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Question 22. Does C(P ), the space of all continuous functions from the pseudoarc into itself, contain any nondegenerate connected sets other than collections of constant maps? The above two questions are also of interest for nondegenerate hereditarily indecomposable continua in general, not just for the pseudo-arc. The Menger universal curve M has quite different local structure from an hereditarily indecomposable continuum. For it, the complexity of this local structure has allowed Brechner [12] to show that H(M ), the topological group of all self-homeomorphisms of M, is totally disconnected and Oversteegen and Tymchatyn [67] to show that H(M ) is one-dimensional.

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Question 23. What is the dimension of H(P ), the topological group of all selfhomeomorphisms of the pseudo-arc P ? There is one aspect in which H(P ) differs from H(M ). Using essential maps onto simple closed curves in M, Brechner [12] has shown that given any two distinct self-homeomorphisms f and g of the Menger curve M there exists > 0 and a separation of H(M ) into sets A and B, with f ∈ A, g ∈ B, and dist(A, B) > , where distance is measured by the sup metric. For the pseudo-arc P [49], given any homeomorphism h : P → P and any > 0, there exist self-homeomorphisms h1 , h2 , . . . , hn of the pseudo-arc such that h = hn ◦· · ·◦h2 ◦h1 and dist(hi , idP ) < for each 1 ≤ i ≤ n. No method of classifying “essential” maps or homeomorphism has been identified which is inherent to the structure of hereditarily indecomposable continua, though the composant structure of such continua and their subcontinua imposes strong constraints on continuous families of maps or homeomorphisms.

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It would also be of interest to know more about the algebraic structure of the topological group H(P ) of all self-homeomorphisms of the pseudo-arc. This author [50] has shown that every inverse limit of finite solvable groups acts effectively on the pseudo-arc. Thus, for every positive integer n, there exists a period n homeomorphism of the pseudo-arc [47]. Though the pseudo-arc is chainable, in the construction of periodic homeomorphisms it is convenient to view it as an inverse limit of n-ods, with the period n homeomorphisms being realized as the restrictions of period n rotations of the plane. The pseudo-arc also admits [44] effective p-adic Cantor group actions. The smallest group not known to act effectively on the pseudo-arc is A5 , the alternating group on 5 symbols, a simple group of order 60. Question 24. Does every compact zero-dimensional topological group act effectively on the pseudo-arc?

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The following question is due to Brechner [13]. It is also of interest without the assumption that the homeomorphism is periodic, or with an assumption of nth roots for n ≥ 2. Question 25. Does each periodic homeomorphism h of the pseudo-arc have a square root, i.e., a homeomorphism g such that g ◦ g = h? While there are limited results on families of homeomorphisms of the pseudoarc, it has more often been possible to determine if there exists a homeomorphism of the pseudo-arc with specific properties. If none of the properties involves extendability to a homeomorphism of the plane or other Euclidean space and if the chainability of the pseudo-arc and the relations between composants of the pseudoarc or of its subcontinua are not inconsistent with the desired set of properties, a homeomorphism with the desired properties can usually be shown to exist. While many of the questions posed so far are specifically phrased in terms of the pseudo-arc, the corresponding versions for such continua as the pseudo-circle, pseudo-solenoids or other continua all of whose nondegenerate proper subcontinua are pseudo-arcs are also of interest. Q-like continua A continuum X is Q-like for the polyhedron Q if, for each > 0, there exists a continuous surjection f : X → Q such that diam(f −1 (q)) < for each q ∈ Q. We have been using the term chainable, which is equivalent to arc-like. It is known that the pseudo-arc P is Q-like for every nondegenerate connected polyhedron Q. It is also true, but not so often recognized, than any nondegenerate chainable continuum which is either indecomposable or 2-indecomposable is Q-like for every nondegenerate connected polyhedron Q. (A continuum is 2-indecomposable if it is the union of two proper subcontinua each of which is indecomposable.) Ingram [27] has constructed an atriodic simple triod-like continuum which is not chainable (arc-like). He [28] has constructed a family of c = 2ℵ0 distinct such continua such that the members of the family can be embedded disjointly

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in the plane. It is a classic result of Moore [64] that there does not exist an uncountable family of disjoint triods in the plane. Ingram’s family also has the property that any continuum can be continuously mapped onto at most countably many members of the family. Ingram has constructed such atriodic simple triod-like nonchainable continua with the property that every nondegenerate proper subcontinuum is an arc, as well as ones such that every nondegenerate proper subcontinuum is a pseudoarc [29, 30]. Minc [58] has constructed an atriodic simple 4-od-like continuum which is not simple triod-like. His example also has the property that every nondegenerate proper subcontinuum is an arc and is obtained from an inverse limit of simple 4-ods with simplicial bonding maps and the same single step bonding map each time. 685?

Question 26. Does there, for every n ≥ 2, exist an atriodic simple (n+1)-od-like continuum which is not simple n-od-like? such an example which is planar? such an example with the property that every nondegenerate proper subcontinuum is an arc? a pseudo-arc? One can place a partial order on the family of nondegenerate connected topological graphs where G1 ≤ G2 if there is a continuous surjection f : G2 → G1 with each nondegenerate f −1 (g), g ∈ G1 , being a connected subgraph of G2 . Under this partial order, if G1 ≤ G2 , then every G1 -like continuum is G2 -like. The arc and the simple closed curve are the minimals elements in this partial order.

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Question 27. If G is a nondegenerate connected topological graph, does there exist an atriodic G-like continuum which is not H-like for any graph H < G?

Hyperspaces Kelley [31] has shown that both indecomposability and hereditary indecomposability can be characterized in terms of the hyperspace C(X) of a nondegenerate continuum X. The nondegenerate continuum X is indecomposable if and only if C(X) \ {X} is not arcwise connected. The nondegenerate continuum X is hereditarily indecomposable if and only if C(X) is uniquely arcwise connected. Eberhart and Nadler [21] have shown that for every nondegenerate hereditarily indecomposable continuum X the hyperspace C(X) is either two-dimensional or infinite-dimensional. This author [41] has shown that every nondegenerate hereditarily indecomposable continuum is the open, monotone image of a onedimensional hereditarily indecomposable continuum. Thus, there exist one-dimensional hereditarily indecomposable continua with infinite-dimensional hyperspaces. Levin and Sternfeld [40] have shown that every continuum of dimension two or greater has infinite-dimensional hyperspace and Levin [39] has shown that every two-dimensional continuum contains a one-dimensional subcontinuum with infinite-dimensional hyperspace.

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Question 28. What is a characterization, either algebraic or topological, of the one-dimensional hereditarily indecomposable continua with infinite-dimensional hyperspaces?

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The dimension-raising maps constructed by the author involve “collapsing holes” to raise dimension. There must be many such “holes.” Rogers [75] has shown that if X is a one-dimensional hereditarily indecomposable continuum with ˇ finitely generated first Cech cohomology then C(X) is two-dimensional. Tymchatyn [93] has shown that if X is a nondegenerate hereditarily indecomposable plane continuum then C(X) can be embedded in R3 . Krasinkiewicz [33] has shown that if X is a nondegenerate hereditarily indecomposable continuum then C(X) doe not contain any subcontinuum homeomorphic to Y × [0, 1] for a nondegenerate continuum Y. He has asked the following question. Question 29. If X is an hereditarily indecomposable continuum, can C(X) ever contain the product of two nondegenerate continua?

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Hereditarily indecomposable continua are characterized by their hyperspaces of subcontinua in the sense that X and Y are homeomorphic if and only if C(X) and C(Y ) are homeomorphic. The composant structure of an hereditarily indecomposable continuum and of its subcontinua imposes a branching which occurs everywhere in the hyperspace. Ball, Hagler and Sternfeld [1] have shown that, while distinct hereditarily indecomposable continua have distinct hyperspaces of subcontinua, there is a natural ultrametric which can be placed on the hyperspace C(X) of an hereditarily indecomposable continuum which yields a topology finer than that normally placed on C(X) by the Hausdorff metric and with the property that C(X) and C(Y ) are homeomorphic and in fact isometric for any nondegenerate hereditarily indecomposable continua X and Y. Dimensions greater than one Hereditarily indecomposable continua of dimension greater than one have complex structure. For example, it is known from results of Mason, Walsh and Wilson [56] that no such continuum can be P -like for any polyhedron P, i.e., if such a continuum is expressed as an inverse limit of polyhedra, the factor spaces in the inverse sequence must get increasingly complex. The following questions are due to Krasinkiewicz [34]. Question 30. Suppose X is an hereditarily indecomposable continuum such that dim(X) = n ≥ 2. Does there exist an essential map from X onto the n-dimensional sphere S n ? (By a result of Krasinkiewicz, if dim(X) > n, then there does exist such an essential map onto S n .)

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Question 31. If X is an hereditarily indecomposable continuum and A is a subcontinuum of X, is it true that Sh(A) ≤ Sh(X)? (Sh(A) denotes the shape of A.)

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Question 32. If X is an hereditarily indecomposable continuum and A is a subcontinuum of X, is it true that A is a shape retract of X? Under what conditions is A a(n open) retract of X?

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Question 33. If X is an hereditarily indecomposable continuum with dim(X) ≥ 2 does there exist a continuous surjection from X onto a nontrivial solenoid? onto a nontrivial pseudo-solenoid? The pseudo-arc has a very rich collection of self-homeomorphisms. There are other hereditarily indecomposable continua with very few homeomorphisms. Cook [17] has constructed one-dimensional hereditarily indecomposable continua with very few self-maps and no nonidentity self-homeomorphisms. One example has the property that any map between subcontinua is either a constant or a retract onto subcontinua, while another has the stronger property that any map between subcontinua is either a constant or the identity map of a subcontinuum onto itself. Pol [69] has constructed, for every positive integer n, hereditarily indecomposable continua of arbitrary positive dimension, whose groups of self-homeomorphisms are cyclic groups of order n. She [70] has recently constructed, for every positive integer n, an hereditarily indecomposable one-dimensional continuum Xn with exactly n continuous self-surjections each of which is a homeomorphism and such that Xn admits an atomic map onto the pseudo-arc. (A mapping f : X → Y between continua is atomic if, for each subcontinuum K of X such that f (K) is nondegenerate, f −1 (f (K)) = K.)

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Question 34. Which finite groups are the complete homeomorphism groups of hereditarily indecomposable continua? For each such group, does there exist an hereditarily indecomposable continuum with that group as its homeomorphism group such that every continuous self-surjection of the continuum is a homeomorphism? such a continuum of arbitrary positive dimension? Renska [73] has constructed for every m = 2, 3, . . . , ∞ an m-dimensional hereditarily indecomposable Cantor manifold Ym whose only self-homeomorphism is the identity. Pol [68] has constructed for every such m an m-dimensional hereditarily indecomposable continuum whose only continuous self-surjection is the identity. For each such m she has constructed c = 2ℵ0 such continua which are pairwise incomparable by continuous maps.

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Question 35. Does there for each m = 2, 3, . . . , ∞ exist an m-dimensional hereditarily indecomposable Cantor manifold whose only continuous self-surjection is the identity? Non-metric continua If x is a point of the nondegenerate continuum X, the composant Cx of X corresponding to the point x is the union of all proper subcontinua of X containing the point x. If X is a nondegenerate decomposable continuum, then X always has either exactly one composant, in the case when X is not irreducible between any

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pair of points, or exactly three distinct composants, in the case when X is irreducible between some pair of points, with any two of the composants intersecting. This is the case whether X is metrizable or not. For indecomposable continua, the case is quite different. A nondegenerate indecomposable metric continuum always has c = 2ℵ0 distinct composants, which form a partition of the continuum with each composant being a dense first category set in the continuum. Bellamy [3] has constructed a non-metric indecomposable continuum with exactly two composants and, by identifying a point in each composant [4], a non-metric indecomposable continuum with only one composant. Smith [87] has constructed a non-metric hereditarily indecomposable continuum with exactly two composants. Bellamy’s identification to produce a single composant produces a decomposable subcontinuum and so cannot be used to obtain an hereditarily indecomposable continuum. Question 36. Does there exist a non-metric hereditarily indecomposable continuum with only one composant?

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Smith [85] has also shown that for every infinite cardinal α there is an indecomposable continuum with exactly 2α composants. ˇ If H = [0, 1) and H∗ is the Stone–Cech remainder H∗ = β(H) \ H, then ∗ H is an indecomposable continuum. However, in set theory determined by ZFC alone it is not possible to determine the number of composants of H∗ . There are consistency results by Rudin, Blass, Banakh, Mildenberger and Shelah showing that the number of composants of H ∗ can be 1, 2 or 2c . Banakh and Blass [2] have shown that the number of composants of H∗ must be either finite or 2c , but it is not known that if the number of composants is finite it must be one or two. Question 37. If X is an indecomposable non-metric continuum with only finitely many composants, must X have at most two composants?

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Question 38. If X is an indecomposable non-metric continuum with infinitely many composants, must the number of composants of X be 2α for some infinite cardinal α?

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By considering inverse limits of pseudo-arcs indexed by ω1 , Smith [89] has constructed a non-metric hereditarily indecomposable homogeneous hereditarily equivalent continuum. Thus this continuum shares many of the properties of the metric pseudo-arc. He has also constructed an inverse limit of pseudo-arcs indexed by ω1 which is a non-metric hereditarily indecomposable continuum which is neither homogeneous nor hereditarily equivalent. The first example has c = 2ℵ0 composants, while the second example has only two composants. The following questions are due to Smith. Question 39. Are there non-metric indecomposable hereditarily equivalent continua other than the inverse limit of ω1 pseudo-arcs constructed by Smith?

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Question 40. Are there non-metric homogeneous chainable continua other than the inverse limit of ω1 pseudo-arcs constructed by Smith? In particular, is there an inverse limit on a large set of chainable continua which is homogeneous?

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Question 41. How many topologically distinct continua obtainable as inverse limits of pseudo-arcs indexed by ω1 are there? Bing [8] has shown that every metric continuum of dimension greater than one contains an hereditarily indecomposable continuum. Smith [84, 88, 90] has a number of results and examples for products of non-metric continua and for non-metric continua of various dimensions showing that the analogous situation is not true, or that there may exist indecomposable continua but not hereditarily indecomposable continua. This is an area deserving much further investigation. Conclusion Kuratowski [37] wrote in 1973 about the theory of indecomposable continua: “It is one of the most developed and, to my mind, most beautiful branches of topology. It has attracted the attention of such distinguished mathematicians as P.S. Alexandrov, RH Bing, D. van Dantzig, G.W. Henderson, S. Mazurkiewicz, E.E. Moise, R.L. Moore, and, among the younger generation, C. Hagopian, J. Krasinkiewicz, Rogers and many others. “This is not surprising: by means of indecomposable continua one has succeeded in solving many earlier problems and in opening up new, extraordinarily rich topics. “In particular, great interest has been aroused (and this is even more noteworthy) by hereditarily indecomposable continua, i.e. those whose every subcontinuum is indecomposable (B. Knaster gave the first example of an hereditarily indecomposable continuum in 1922 in “Fund. Math.” 3).” Except that individuals are not as young as they once were, this is just as true more than 30 years later as it was when Kuratowski wrote it and indecomposable continua continue to richly reward anyone willing to investigate them. References [1] R. N. Ball, J. N. Hagler, and Y. Sternfeld, The structure of atoms (hereditarily indecomposable continua), Fund. Math. 156 (1998), no. 3, 261–278. [2] T. Banakh and A. Blass, The number of near-coherence classes of ultrafilters is either finite or 2c , 2005, Preprint. To appear in the proceedings of the set theory year at CRM (Barcelona). [3] D. P. Bellamy, A non-metric indecomposable continuum, Duke Math. J. 38 (1971), 15–20. [4] D. P. Bellamy, Indecomposable continua with one and two composants, Fund. Math. 101 (1978), no. 2, 129–134. [5] D. P. Bellamy and J. A. Kennedy, Factorwise rigidity of products of pseudo-arcs, Topology Appl. 24 (1986), 197–205. [6] D. P. Bellamy and J. M. Lysko, Factorwise rigidity of the product of two pseudo-arcs, Topology Proc. 8 (1983), no. 1, 21–27.

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[7] R H Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. [8] R H Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267–273. [9] R H Bing, Snake-like continua, Duke Math. J. 18 (1951), 653–663. [10] R H Bing, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345–346. [11] R H Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119–132. [12] B. L. Brechner, On the dimensions of certain spaces of homeomorphisms, Trans. Amer. Math. Soc. 121 (1966), 516–548. [13] B. L. Brechner, Problem 8 in Continuum Theory Problems, Topology Proc. 8 (1983), 361– 394. [14] C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math. 9 (1959), 653– 659. [15] J. J. Charatonik and T. Ma´ ckowiak, Confluent and related mappings on arc-like continua— an application to homogeneity, Topology Appl. 23 (1986), no. 1, 29–39. [16] H. Cook, Weakly confluent mappings and dimension, 1977 Spring Topology Conference Abstracts, Baton Rouge (1977) 31. [17] H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. [18] H. Cook, Tree-likeness of hereditarily equivalent continua, Fund. Math. 68 (1970), 203–205. [19] H. Cook, W. T. Ingram, and A. Lelek, A list of problems known as Houston problem book, Continua (Cincinnati, OH, 1994), Lecture Notes in Pure and Appl. Math., vol. 170, Marcel Dekker Inc., New York, 1995, pp. 365–398. [20] J. L. Cornette, Retracts of the pseudo-arc, Colloq. Math. 19 (1968), 235–239. [21] C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys. 19 (1971), 1027–1034. [22] L. Fearnley, Characterizations of the continuous images of the pseudo-arc, Trans. Amer. Math. Soc. 111 (1964), 380–399. [23] C. L. Hagopian, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), no. 2, 339–350 (1977). [24] C. L. Hagopian, Atriodic homogeneous continua, Pacific J. Math. 113 (1984), no. 2, 333–347. [25] C. L. Hagopian and J. T. Rogers, Jr., A classification of homogeneous, circle-like continua, Houston J. Math. 3 (1977), no. 4, 471–474. [26] G. W. Henderson, Proof that every compact decomposable continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc, Ann. of Math. (2) 72 (1960), 421–428. [27] W. T. Ingram, An atriodic tree-like continuum with positive span, Fund. Math. 77 (1972), no. 2, 99–107. [28] W. T. Ingram, An uncountable collection of mutually exclusive planar atriodic tree-like continua with positive span, Fund. Math. 85 (1974), no. 1, 73–78. [29] W. T. Ingram, Hereditarily indecomposable tree-like continua, Fund. Math. 103 (1979), no. 1, 61–64. [30] W. T. Ingram, Hereditarily indecomposable tree-like continua. II, Fund. Math. 111 (1981), no. 2, 95–106. [31] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36. [32] B. Knaster, Un continu dont tout sous-continu est ind´ ecomposable, Fund. Math. 3 (1922), 247–286. [33] J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84 (1974), no. 3, 175–186. [34] J. Krasinkiewicz, Mapping properties of hereditarily indecomposable continua, Houston J. Math. 8 (1982), no. 4, 507–516.

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[64] R. L. Moore, Concerning triodic continua in the plane, Fund. Math. 13 (1928), 261–263. [65] L. G. Oversteegen and E. D. Tymchatyn, Plane strips and the span of continua. I, Houston J. Math. 8 (1982), no. 1, 129–142. [66] L. G. Oversteegen and E. D. Tymchatyn, On span and weakly chainable continua, Fund. Math. 122 (1984), no. 2, 159–174. [67] L. G. Oversteegen and E. D. Tymchatyn, On the dimension of certain totally disconnected spaces, Proc. Amer. Math. Soc. 122 (1994), no. 3, 885–891. [68] E. Pol, The non-existence of common models for some classes of higher-dimensional hereditarily indecomposable continua, Preprint. [69] E. Pol, Hereditarily indecomposable continua with exactly n autohomeomorphisms, Colloq. Math. 94 (2002), no. 2, 225–234. [70] E. Pol, Hereditarily indecomposable continua with finitely many continuous surjections, Bol. Soc. Mat. Mexicana (3) 11 (2005), no. 1, 139–147. [71] J. R. Prajs, On open homogeneity of closed balls, 1992, Preprint. [72] J. R. Prajs, Plane continua homogeneous with respect to light open mappings, 1993, Preprint. [73] M. Re´ nska, Rigid hereditarily indecomposable continua, Topology Appl. 126 (2002), no. 1–2, 145–152. [74] J. T. Rogers, Jr., Solenoids of pseudo-arcs, Houston J. Math. 3 (1977), no. 4, 531–537. [75] J. T. Rogers, Jr., Weakly confluent mappings and finitely-generated cohomology, Proc. Amer. Math. Soc. 78 (1980), no. 3, 436–438. [76] J. T. Rogers, Jr., Homogeneous, separating plane continua are decomposable, Michigan Math. J. 28 (1981), no. 3, 317–322. [77] J. T. Rogers, Jr., Homogeneous hereditarily indecomposable continua are tree-like, Houston J. Math. 8 (1982), no. 3, 421–428. [78] J. T. Rogers, Jr., Cell-like decompositions of homogeneous continua, Proc. Amer. Math. Soc. 87 (1983), no. 2, 375–377. [79] J. T. Rogers, Jr., Homogeneous continua, Topology Proc. 8 (1983), no. 1, 213–233. [80] J. T. Rogers, Jr., Classifying homogeneous continua, Topology Appl. 44 (1992), 341–352. [81] I. Rosenholtz, Open maps of chainable continua, Proc. Amer. Math. Soc. 42 (1974), 258– 264. [82] C. R. Seaquist, Monotone homogeneity and planar manifolds with boundary, Topology Appl. 81 (1997), no. 1, 47–53. [83] C. R. Seaquist, Monotone open homogeneity of Sierpi´ nski curve, Topology Appl. 93 (1999), no. 2, 91–112. [84] M. Smith, On non-metric indecomposable and hereditarily indecomposable subcontinua of products of long Hausdorff arcs, Preprint. [85] M. Smith, Generating large indecomposable continua, Pacific J. Math. 62 (1976), no. 2, 587–593. [86] M. Smith, Large indecomposable continua with only one composant, Pacific J. Math. 86 (1980), no. 2, 593–600. [87] M. Smith, A hereditarily indecomposable Hausdorff continuum with exactly two composants, Topology Proc. 9 (1984), no. 1, 123–143. [88] M. Smith, Hausdorff hypercubes which do not contain arcless continua, Proc. Amer. Math. Soc. 95 (1985), no. 1, 109–114. [89] M. Smith, On nonmetric pseudo-arcs, Topology Proc. 10 (1985), no. 2, 385–397. [90] M. Smith, Continua arbitrary products of which do not contain nondegenerate hereditarily indecomposable continua, Topology Proc. 13 (1988), no. 1, 137–160. [91] M. Sobolewski, Pseudoarc is not pseudocontractible, 1991, Preprint. [92] J. A. Toledo, Inducible periodic homeomorphisms of tree-like continua, Trans. Amer. Math. Soc. 282 (1984), no. 1, 77–108. [93] E. D. Tymchatyn, Hyperspaces of hereditarily indecomposable plane continua, Proc. Amer. Math. Soc. 56 (1976), 300–302.


Open problems on dendroids Veronica Mart´ınez-de-la-Vega and Jorge M. Mart´ınez-Montejano Dedicated to the memory of our friend, colleague and teacher, Professor Janusz J. Charatonik. 1. Introduction A continuum is a compact, connected, metric space. A dendroid is an arcwise connected and hereditarily unicoherent continuum. A dendrite is defined as a locally connected dendroid. Dendroids were defined by B. Knaster in the Topology Seminar in Wroclaw in the late 1950s. One of the most assiduous participants of this Seminar was Prof. J.J. Charatonik, who wrote his Doctoral Dissertation and over fifty papers on dendroids. Many of his doctoral students, such as S.T. Czuba, T. Maćkowiak, P. Krupski and J. Prajs, also have done many contributions to this field. Even though dendroids are one-dimensional and most of them can be geometrically realized, there are many properties and intrinsic characterizations of them which are still unknown. The purpose of this paper is to give a survey of some results and open problems in this interesting area of Continuum Theory. 2. The problem We begin with a problem that many authors consider as one of the most importance in the study of dendroids. Before posing the problem it is interesting to note that in the early 1960s, B. Knaster saw dendroids as those continua that for which for every ε > 0 there exists a tree T and an ε-retraction r : X → T (an ε-retraction is a retraction such that diam(r −1 (t)) < ε for every t ∈ T ). The contemporary definition of dendroids (the one given above) was formulated in a more convenient way. Problem 1. Let X be a dendroid and ε > 0. Are there a subtree T of X and an ε-retraction from X onto T ?

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Let us note that Fugate in [26, 27] has some partial positive answers to this problem and that a positive answer to it would give an affirmative answer to a variety of other problems about dendroids (see, e.g., Problem 27). 3. Mappings on dendrites Characterize dendrites among dendroids is one of the eldest problems in the study of dendroids, over 60 characterizations of dendrites can be found in [10]. Also in [10] a survey of some open problems is made. Problem 2 ([10, Problem 2.14]). Characterize all dendrites X having the property that every open image of X is homeomorphic to X.

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Problem 3 ([10, Question 3.6]). Does every monotonely homogeneous dendrite (Definition 3.8) contain a homeomorphic copy of the dendrite L0 ?

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§35. Mart´ınez-de-la-Vega and Mart´ınez-Montejano, Open problems on dendroids

A topological space X is said to be chaotic (strongly chaotic) if for any two distinct points p and q of X there exists open neighborhoods U of p and V of q such that no open subset of U is homeomorphic to any open subset (subset) of V ; and rigid (strongly rigid) if the only homeomorphism of X onto (into) X is the identity map. 704–705?

Problem 4 ([10, Problem 4.5]). Give any structural characterization of (strongly) chaotic and of (strongly) rigid dendrites. Consider the following conditions (ω0 ) and (ω). (ω0 ) For every compact space Y , for every light open mapping f : Y → f (Y ) with X ⊂ f (Y ) and for every point y0 ∈ f −1 (X) ⊂ Y there exists a homeomorphic copy X 0 of X in Y with y0 ∈ X 0 such that the restriction f |X 0 : X 0 → f (X 0 ) = X is a homeomorphism. (ω) For every compact space Y, for every light open mapping f : Y → f (Y ) with X ⊂ f (Y ) there exists a homeomorphic copy X 0 of X in Y such that the restriction f |X 0 : X 0 → f (X 0 ) = X is a homeomorphism. Now consider conditions (ω0 (M)) and (ω0 (M)) regarding a continuum X and a class M of light mappings. Which can be defined replacing the phrase “light open mapping f : Y → f (Y )” by “light mapping f : Y → f (Y ) in M.” Note that if O stands for the class of open mappings, then (ω0 (O)) and (ω(O)) coincide with conditions (ω0 ) and (ω). One can also consider conditions (γ0 ) and (γ) obtained from (ω0 ) and (ω), respectively, by replacing the phrase “every compact space Y ” with “every continuum Y ”, and define condition (δ) X is a dendrite. In [18, Statement 1] it is shown why conditions (δ), (ω0 ), (ω), (γ0 ) and (γ) are equivalent. So the following problem is posed.

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Problem 5 ([18, Problem 2]). Does the equivalence in [18, Statement 1] remain true if we replace openness of the light mapping f by a less restrictive condition? In other words, for what (larger) classes M of light mappings are conditions (δ), (ω0 ), (ω), (γ0 ) and (γ) equivalent? Also by [18, Observation 4] if the class M contains the class O of open mappings, and if, for a continuum X, implication (δ) ⇒ (ω0 (M)) holds, then all the conditions (δ), (ω0 ), (ω), (ω0 (M)) and (ω(M)) are equivalent. Thus Problem 5 reduces to:

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Problem 6 ([18, Problem 5]). For what classes M of mappings containing the class O does each dendrite X satisfy condition (ω0 (M)) (i.e., the implication (δ) ⇒ (ω0 (M)) holds)? In [7] the following result of K. Omiljanowski is proved. Theorem 3.1. Let a dendrite Y be such that all ramification points of Y are of order 3 and the set R(Y ) of all ramification points of Y is discrete. If a dendrite X can be mapped onto Y under a monotone mapping, then X contains a homeomorphic copy of Y .

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In [2] J.J. Charatonik proved the following theorem. Theorem 3.2. Let D be a dendrite. For every compact space X and for every light open surjective mapping f : X → Y with D ⊂ Y there is a homeomorphic copy D0 of D in X such that the restriction f |D0 : D0 → f (D0 ) is a homeomorphism. The inverse implication of Theorem 3.2 was proved in [11, Corollary 10] and [18, Theorem 16]. It is interesting to ask if Theorem 3.1 can be reversed. So the natural problem is. Problem 7 ([2, Problem 1.3]). Characterize all dendrites Y having the property that if a dendrite X can be mapped onto Y under a monotone mapping, then X contains a homeomorphic copy of Y .

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Using the notation and all the classes of mappings defined in [2], the following Corollary is proved. Corollary 3.3 ([2, Corollary 3.2]). Let a continuum Y satisfy the conditions of Theorem 3.1. If a dendrite X can be mapped onto Y under a mapping that belongs to one of the classes of mappings OM, C, LocC, QM, WM, then X contains a homeomorphic copy of Y . Problem 8 ([2, Question 3.3]). Let a continuum Y satisfy conditions of Theorem 3.1 and X a dendrite that can be mapped onto Y under a semi-confluent mapping. Must then X contain a homeomorphic copy of Y ?

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Notice that Theorem 3.1 cannot be extended to continua X that contain simple closed curves, not even if X is locally connected or X is a linear graph [53, Ch. X, §3, Ex., p. 189]). Using all the above ideas one of the natural questions is: Problem 9 ([2, Question 3.7]). Let a continuum Y satisfy conditions of Theorem 3.1 and let a continuum X be such that if X can be mapped onto Y under a monotone mapping, then X contains a homeomorphic copy of Y . Must then X be a dendrite? If not under what additional assumptions does the implication hold?

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Another important and interesting question related to Theorem 3.1 is if the implication in the result can be reversed. A partial answer is given. Theorem 3.4 ([2, Theorem 4.1]). Let a dendrite Y have the property that for each dendrite X if X can be mapped onto Y under a monotone mapping, then X contains a homeomorphic copy of Y . Then either Y is an arc or all the ramification points of Y are of order 3. And so the following question arises. Problem 10 ([2, Question 4.2]). Let a dendrite Y have the same property as in Theorem 3.4. Must then Y either be an arc or have the set R(Y ) of ramification points discrete? Given a space X and a map f : X → X. A point x of X is said to be fixed if f (x) = x, periodic (of period n) provided that there is n ∈ N such that f n (x) = x

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(and f k (x) 6= x for k < n), recurrent, provided that for each open set U containing x there is n ∈ N such that f n (x) ∈ U , eventually periodic of period n provided that there exists m ∈ N ∪ {0} such that f m (x) is a periodic point of f of period n, eventually periodic provided that there is n ∈ N such that x is an eventually periodic point of period n ∈ N for f and a non-wandering point of f provided that for any open set U containing x there exists y ∈ U and n ∈ N such that f n (y) ∈ U . For a mapping f : X → X the sets of fixed, periodic, recurrent, eventually periodic and non-wandering points of f will be denoted by F (f ), P (f ), R(f ), EP (f ) and Ω(f ), respectively. Also a space X is said to have the periodic-recurrent property (P R-property) provided that for every mapping f : X → X the equality cl(P (f )) = cl(R(f )) holds, the non-wandering-periodic property (ΩP -property) provided that for every mapping f : X → X the equality Ω(f ) = P (f ) holds and the non-wanderingeventually periodic property (ΩEP -property) provided that for every mapping f : X → X the inclusion Ω(f ) ⊂ clX (EP (f )) is satisfied. In [16, Proposition 2.11] it is proved that a dendrite X has the P R-property if and only if X does not contain any copy of the Gehman dendrite; and the following problem is posed. 712?

Problem 11 ([16, Problem 2.14]). Give an internal (i.e., structural) characterization of dendrites with ΩP -property. In [16, Corollary 3.6] it is proved that if X is a tree and f a mapping f : X → X such that Ω(f ) is finite, then card(P (f )) = card(Ω(f )). It is not known if the assumption of the finiteness of the set Ω(f ) is or is not essential. So, the following question is posed.

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Problem 12 ([16, Question 3.7]). Do there exist a tree X and a mapping f : X → X such that Ω(f ) is infinite while P (f ) is finite? In [16, Theorem 4.6] it is proved that if X is a dendrite such that for each mapping f : X → X the equality card(P (f )) = card(Ω(f )) holds, then X is a tree. So, it is asked if the converse is true.

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Problem 13 ([16, Question 4.7]). Is it true that for each tree X the assertion that for each mapping f : X → X the equality card(P (f )) = card(Ω(f )) holds? Given a dendroid X, we define E(X) as the set of endpoints, O(X) the set of ordinary points and R(X) the set of ramification points of X. In [7, Proposition 4.7] J.J. Charatonik proved the following. Proposition 3.5. Let x and y be any two points of the standard universal dendrite X = Dm for m ∈ {3, 4, . . . , ω}. Then there is a homeomorphism h : X → X such that h(x) = y if and only if one of the following conditions is satisfied: x, y ∈ E(X); x, y ∈ O(X); x, y ∈ R(X). To this respect the following problem remains unsolved.

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Problem 14 ([7, Question 4.9]). What dendrites X have the property that for each two points x and y of X there exists a homeomorphism h : X → X with

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h(x) = y if and only if both these points are either end points, or ordinary points or ramification points of X? It is known that: Theorem 3.6 ([7, Corollary 5.5]). Let Dm be the standard universal dendrite of order m ∈ {3, 4, . . . , ω}. Then each monotone surjection of Dm onto itself is a near homeomorphism if and only if m = 3. A map f : X → Y is a near homeomorphism provided that for ε > 0 there exists a homeomorphism h : X → Y such that sup{d(f (x), h(x) : x ∈ X} < ε. The following problem is still open. Problem 15 ([7, Problem 5.1]). What dendrites X have the property that each monotone mapping of X onto itself is a near homeomorphism?

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Let M be a class of mappings. Two continua X and Y are said to be equivalent with respect to M if there are a mapping in M from X onto Y and a mapping in M from Y onto X. A class M of mappings is said to be neat if all homeomorphisms are in M and the composition of any two mappings in M is also in M. Therefore, if a neat class M of mappings is given, then a family of continua is decomposed into disjoint equivalence classes in the sense that two continua belong to the same class provided that they are equivalent with respect to M. A continuum is said to be isolated with respect to M provided the above mentioned class to which X belongs consists of X only. In [7, Theorems 6.7 and 6.14] it is shown that universal dendrites are not isolated with respect to monotone mappings. The following problem remain open. Problem 16 ([7, Problem 6.1]). Find all dendrites which are isolated with respect to monotone mappings.

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About the previous problem the authors have the following conjecture. Conjecture 3.7. A dendrite X is isolated with respect to monotone mappings if and only if |R(X)| < ∞.

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Definition 3.8. Let M be a class of mappings. A continuum X is homogeneous with respect to M provided that for every two points p and q of X there is a surjective mapping f : X → X such that f (p) = q and f ∈ M. In [7, Theorem 7.1] it is proved that any standard universal dendrite Dm of order m ∈ {3, 4, . . . , ω} is homogeneous with respect to monotone mappings. After this it is observed that each m-od is an example of a dendrite which is not homogeneous with respect to confluent, and therefore to monotone, mappings. Then the following problem is posed. Problem 17 ([7, Question 7.2]). What dendrites are homogeneous with respect to monotone mappings?

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4. Maps onto dendroids In [32, Theorem 2], J. Heath and V. Nall proved the following. Theorem 4.1. There does not exist a (exactly) 2-to-1 map from a hereditarily decomposable continuum onto a dendroid. They asked the following. 720?

Problem 18 ([32, p. 288]). Is there an indecomposable continuum I that admits a map onto a dendroid X such that the inverse of each point in the range contains at most two points? A negative answer to Problem 18 would have strengthened Theorem 4.1. P. Minc has shown that this is the case in some special situations and he says that it would be interesting to partially answer Problem 18 for the case of chainable continua. He proved the next two theorems. Theorem 4.2 ([44, Corollary 3.12]). Let f be a map of an indecomposable continuum Y into a plane dendroid P . Then there is a point p ∈ P such that f −1 (p) is uncountable. Theorem 4.3 ([44, Corollary 3.9 and Remark 3.10]). Let K be either any Knaster type continuum or any solenoid. Suppose that f is a map of K onto an arbitrary dendroid X. Then there is a point x ∈ X such that f −1 (x) consists of at least three points. Also, P. Minc pointed out the following: Theorem 4.3 shows that some indecomposable continua do not admit 2-or-1-to-1 maps on dendroids. On the other hand, it is easy to construct such maps from many standard examples of chainable hereditarily decomposable continua. So he posed the next problem.

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Problem 19 ([45, p. 289]). Is it true that a chainable continuum is hereditarily decomposable if and only if it admits a 2-or-1-to-1 map onto a dendroid? Let us note that in [45, Theorem 1.1] he proved that every chainable continuum can be mapped into a dendroid in such a way that all point-inverses consist of at most three points. 5. Contractibility The symbols Lsup , Linf and Lt mean the upper limit, the lower limit and the topological limit. A dendroid X is said to be: (a) smooth, (b) semi-smooth, (c) weakly smooth, if there exists a point p ∈ X such that for every a ∈ X and each convergent sequence {an }n∈N ⊂ X, with an → a we have: (a) Lt pan = pa, (b) Lsup pan is an arc, (c) Linf pan = pb for some b ∈ X. A dendroid X is said to be pointwise smooth if for each x ∈ X there exists a point p(x) ∈ X such that for each convergent sequence xn convergent to a point a, the sequence of arcs p(x)an is convergent and Lt pan = pa. The next theorem is well known. Theorem 5.1 ([5, 6]). Every contractible one-dimensional continuum is a dendroid.

Contractibility

325

It is known that the inverse implication is not true. The main problem related to contractibility of dendroids is the following. Problem 20 ([8, p. 28]). Find a structural characterization of contractible dendroids.

722?

Given a dendroid X, a mapping H : X × [0, 1] → X such that H(x, 0) = x for each point x ∈ X is called a deformation. A nonempty proper subset A of a dendroid X is said to be homotopically fixed provided that for every deformation H : X × [0, 1] → X we have that H(A × [0, 1]) = A. A nonempty subset A of a dendroid X is said to be homotopically steady provided that for every deformation H : X × [0, 1] → X we have that A ⊂ H(X × {1}). Denote by D(X) the family of all T deformations on X. Define, K(X), the kernel of steadiness of X by K(X) = {H(X × {1}) : H ∈ D(X)}. A nonempty proper subcontinuum A of a dendroid X is called an Ri -continuum (where i ∈ {1, 2, 3}) if there exist an open set U containing A and two sequences {Cn1 : n ∈ N} and {Cn2 : n ∈ N} of components in U such that  1 2  Lsup Cn ∩ Lsup Cn for i = 1, 1 2 A = Lt Cn ∩ Lt Cn for i = 2,   1 Linf Cn for i = 3. It is well known that if a dendroid X contains a homotopically fixed subset, then X is not contractible [14, Proposition 1]]) and that each Ri -continuum of a dendroid X (where i ∈ {1, 2, 3}) is a homotopically fixed subset of X [21, Theorem 3]. J.J. Charatonik and A. Illanes proved [15, Theorem 4.3] that each contractible space has empty kernel of steadiness and asked: Problem 21 ([15, Question 4.5]). Does every non-contractible dendroid have nonempty kernel of steadiness?

723?

Also, it is shown that [15, Example 4.7] there is a plane dendroid X and a subcontinuum A of X such that A is an Ri -continuum in X for each i = 1, 2, 3, so it is homotopically fixed, while not homotopically steady. So the following questions arise. Problem 22 ([15, Questions 4.9]). (a) Does the existence of a homotopically fixed subset in a dendroid imply the existence of a homotopically steady subset? (b) What are the interrelations between Ri -continua and homotopically steady subsets of dendroids? More precisely, let an Ri -continuum A (for some i ∈ {1, 2, 3}) be contained in a dendroid X. Must A contain a nonempty homotopically steady subset of X?

724–725?

The following question (asked by W.J. Charatonik) is related to Problem 21. Problem 23 ([15, Question 4.19]). Given a dendroid X with a nondegenerate kernel K(X) of steadiness, is the dendroid X/K(X) always contractible?

726?

326


A point p of a dendroid X is called a Q-point of X provided that there exists a sequence of points pn of X converging to p such that Lsup ppn 6= {p} and if for each n ∈ N the arc pn qn is irreducible between pn and the continuum Lsup ppn , then the sequence of points qn converges also to p. The following problem is open. 727?

Problem 24 ([8, p. 30]). Is it true that if a dendroid has a Q-point, then it is non-contractible? By [23, Corollary 3.10], it is known that if a dendroid is hereditarily contractible, then it is pointwise smooth. So, the following questions arise naturally.

728?

Problem 25 ([23, Question 3.11]). Does pointwise smoothness of dendroids imply their hereditary contractibility?

729?

Problem 26 ([14, Question 13]). Find an intrinsic characterization of hereditarily contractible dendroids. 6. Hyperspaces Given a continuum X, the hyperspace 2X of X is defined by 2X = {A ⊂ X : A is nonempty and closed}. We consider 2X with the Hausdorff metric H. Other hyperspaces considered here are C(X) = {A ∈ 2X : A is connected}, and, for each n ∈ N, Fn (X) = {A ∈ 2X : A has at most n elements}. In [48, Theorem 6.18], S.B. Nadler, Jr. proved that C(X), 2X and Fn (X) have the fixed point property when X is either a smooth dendroid or a fan. Right below the proof, S.B. Nadler, Jr. asks the following.

730–732?

Problem 27. Do C(X), 2X and Fn (X) have the fixed point property when X is a dendroid? We note that a positive answer would follow from an affirmative answer to Problem 1. 7. Property of Kelley A continuum X is said to have the property of Kelley at a point x ∈ X provided that for each sequence of points xn converging to x and for each continuum K in X containing the point x there is a sequence of continua Kn in X with xn ∈ Kn for each n ∈ N and converging to K. A continuum X is said to have the property of Kelley if it has the property at each point x ∈ X. Given a dendroid X, and x ∈ X, we define the Jones function T (x) = {y ∈ X : if there exists A ∈ C(X) such that y ∈ intX (A) then x ∈ A}. A λ-dendroid is a hereditarily unicoherent and hereditarily decomposable continuum. S.T. Czuba proved the following implications.

Retractions

327

Theorem 7.1 ([25, Lemma 2 and Corollary 5]). Let X be a dendroid. Then: X has the property of Kelley =⇒ ∀x, y ∈ X if xy ∩ T (x) 6= {x}, then y ∈ T (x) =⇒ X is smooth =⇒ X is locally connected at some point. In Theorem 7.1, the assumption that X is a dendroid is essential: a λ-dendroid obtained as a compactification of the Cantor fan minus its vertex such that the remainder is an arc has the property of Kelley and is non-smooth (it is not locally connected at any point). Concerning to this J.J. Charatonik asked the following. Problem 28 ([8, Question 5.20]). For what continua X does the property of Kelley imply local connectedness of X at some point?

733?

Fans having the property of Kelley have been characterized in [3, 9]. But there are no known characterizations of dendroids having the property of Kelley. So we have the following. Problem 29. Characterize dendroids having the property of Kelley.

734?

8. Retractions In [12, Theorem 3.1 and Theorem 3.3], J.J. Charatonik et al proved the following. Theorem 8.1. Let X be a one-dimensional continuum. If there is a retraction from C(X) (2X ) onto X, then X is a uniformly arcwise connected dendroid. And, in [28, Theorem 2.9], J.T. Goodykoontz, Jr. showed the following. Theorem 8.2. Every smooth fan X is a deformation retract of 2X . Also, there are known examples of a non-smooth fan X such that there is no retraction from 2X onto X [1, Example 3.7] and of a non-planable smooth dendroid for which there is no retraction from 2X onto X [12, Example 5.52]. So, in [12], J.J. Charatonik et al asked the following. Problem 30. For what smooth dendroids X does there exist a deformation retraction from 2X onto X? Let X be a continuum. A retraction r : 2X → X is said to be associative provided that r(A ∪ B) = r({r(A)} ∪ B) for every A, B ∈ 2X . Let X be a hereditarily unicoherent continuum. A retraction r : 2X → X is said to be internal provided that r(A) ∈ I(A) for each A ∈ 2X , where I(A) denotes the continuum irreducible with respect to containing A. It is known [12, Theorem 3.21] that the Mohler–Nikiel universal smooth dendroid admits an associative retraction and, as we mentioned above, that there is a smooth dendroid which admits no retraction from 2X onto X. So, the following problem arises.

735?

328 736?


Problem 31 ([12, Problem 5.57]). Characterize smooth dendroids X admitting a retraction from 2X onto X. Also, since the Mohler–Nikiel universal smooth dendroid have the property of Kelley, the following problem arises.

737?

Problem 32 (Problem 3.23, [12]). Let X be a dendroid with the property of Kelley. Does there exist a retraction r : 2X → X? 9. Means Given a Hausdorff space X, a mean µ on X is defined as a map µ : X ×X → X such that for each x, y ∈ X we have that µ(x, x) = x and µ(x, y) = µ(y, x). The natural question that comes with the definition is: which spaces, especially metric continua admit a mean? This question has been around for more than half of a century and has been answered for a very small class of spaces. So, the main problem about means is the following.

738–739?

Problem 33 ([12, Problems 5.28 and 5.50]). Characterize metric continua (in particular dendroids) that admit a mean. For a continuum X the existence of a mean µ : X × X → X is equivalent to the existence of a retraction r : F2 (X) → X, where the two concepts are related to each other by the equality µ(x, y) = r({x, y}). In this respect, the existence of a retraction r : 2X → X implies the existence of a mean but it is not known if the inverse implication is true. So we have the following problem.

740?

Problem 34 (Question 5.44, [12]). Does there exist a dendroid X which admits a mean and for which there is no retraction from 2X onto X? Related to the previous problem and Theorem 8.1, we have the next problems.

741?

Problem 35 ([12, Question 5.48]). Let X be a dendroid admitting a mean. Must X be uniformly arcwise connected?

742?

Problem 36 ([12, Question 5.49], compare to Problem 32). Let X be a dendroid with the property of Kelley. Does there exist a mean µ : X × X → X? A mean µ : X × X → X is said to be associative provided that µ(x, µ(y, z)) = µ(µ(x, y), z) for every x, y, z ∈ X. It is known that Theorem 9.1 ([12, Theorem 5.3]). Let X be a locally connected continuum. Then the following conditions are equivalent: (1) X is an absolute retract; (2) there is a retraction r : 2X → X. Moreover, if X is one-dimensional, then each of them is equivalent to any of the following: (3) X is a dendrite;

Selections

329

(4) there exists an associative retraction r : 2X → X; (5) there exists an associative mean µ : X × X → X; (6) there exists a mean µ : X × X → X. About Theorem 9.1, J.J. Charatonik et al asked the following. Problem 37 ([12, Question 5.38]). Assume that X is locally connected. Does (6) imply (5)? Does (5) imply (1)?

743–744?

It is known that there is a smooth dendroid admitting no mean [12, Example 5.52] and that the Mohler–Nikiel universal smooth dendroid admits an associative mean [12, Theorem 3.21 and Proposition 5.16]. So the following problem arises. Problem 38 ([12, Problem 5.56]). Characterize smooth dendroids admitting a mean.

745?

A mean µ on a dendroid X is said to be internal if for each x, y ∈ X, µ(x, y) ∈ xy. M. Bell and S. Watson give an example of a contractible and selectible fan which admits a mean while it does not admit neither an associative mean nor an internal mean [1, Example 4.8]. So they asked the following. Problem 39 ([1, Problem 4.3]). Does a selectible dendroid have a mean? Does a contractible dendroid have a mean?

746–747?

10. Selections A continuous selection for a family H ⊂ 2X is a map s : H → X such that s(A) ∈ A for each A ∈ H. A continuum X is said to be selectible provided that it admits a continuous selection for C(X). In [49], S.B. Nadler, Jr. and L.E. Ward, Jr. proved the following. Theorem 10.1. (1) Every selectible continuum is a dendroid; (2) A locally connected continuum is selectible if and only if it is a dendrite; (3) Each selectible dendroid is a continuous image of the Cantor fan, hence it is uniformly arcwise connected. A selection s : H → X, where H ⊂ 2X , is said to be rigid provided that if A, B ∈ H and s(B) ∈ A ⊂ B, then s(A) = s(B). In [52], L.E. Ward, Jr. showed the following. Theorem 10.2. A continuum X is a smooth dendroid if and only if there exists a rigid selection for C(X). On the other hand, there is an uniformly arcwise connected dendroid which is not selectible and there is a non-smooth dendroid admitting non-rigid selections for its hyperspace of subcontinua [8, Figure 17 and Figure 18 respectively]. In [47], S.B. Nadler, Jr. posed the following problem (which is still open). Problem 40. lectible fans).

Give an internal characterization of selectible dendroids (of se-

748–749?

330


In [41], T. Maćkowiak gave an example of a contractible and non-selectible dendroid, J.J. Charatonik asked for an example with these and additional properties. 750–752?

Problem 41 ([8, Question 8.7]). Is there a contractible and non-selectible dendroid which is (a) planable, (b) hereditarily contractible, (c) a fan? An open selection is a selection that also is an open map. In [42], it is shown that a smooth fan X admits an open selection if and only if X is locally connected. Regarding this topic, the following problems are still unsolved.

753?

Problem 42 ([42, Problem 1]). If X is a finite tree, then does X admit an open selection?

754?

Problem 43 ([42, Problem 2]). Can a non-locally connected dendroid admit an open selection? Let D be a dendrite and Σ(D) the space of selections of D. Trying to give new tools to solve Problem 40, J.E. McParland proved that for each dendrite D, the space Σ(D) (a) is not compact [43, Theorem 3.9], (b) is nowhere dense in DC(D) [43, Theorem 3.10], (c) is not dense in DC(D) [43, Theorem 3.11] and (d) is not an arc [43, Theorem 4.3]. For us it is natural to present the following problem.

755?

Problem 44. Give a wider variety of properties with which Σ(D) is endowed. In particular, is Σ(D) homeomorphic to `2 ? 11. Smooth dendroids S.T. Czuba showed some relations among the different types of smoothness (see definitions of Section 5) and proved: Theorem 11.1 ([20, Theorem 1]). A fan is pointwise smooth if and only if it is smooth. Theorem 11.2 ([23, Theorem 4.6]). If a dendroid is pointwise smooth and weakly smooth, then it is also semi-smooth. Corollary 11.3 ([23, Corollary 4.9]). If a dendroid X is pointwise smooth and semi-smooth, then it is also weakly smooth. Consider the following definitions. Let T be a property and A a class of continua. T is finite (countable) in the class A if there is a finite (countable) set F ⊂ A such that a member X of A has property T if and only if X contains a homeomorphic copy of some member of F. A class A has a common model M under continuous mapping if there is a continuum M belonging to A with the property that every member of A is a continuous image of M . A class A has a universal element U , if there is a a continuum U belonging to A with property that every member A can be homeomorphically embedded into U . Now consider the following classes of continua: (a) dendroids, (b) fans, (c) smooth dendroids, (d) smooth fans, (e) semi-smooth dendroids, (f) semi-smooth

Planability

331

fans, (g) weakly smooth dendroids, (h) weakly smooth fans, (i) pointwise smooth dendroids, (j) uniformly arcwise connected dendroids, (k) uniformly arcwise connected fans. Some of the questions which remain unanswered are. Problem 45. Does there exist a common model for the classes (a), (b), (e) and (f )?

756?

In [13, Theorem 11] and [36] it is shown that classes (c), (d), (j), (k) have a common model. Problem 46. Does there exist a universal element for the classes (b), (e), (f ), (g), (h), (i), (j) and (k)?

757?

A universal element is known for classes (c) [30, 46] and (d) [13, Theorem 10]. Class (a) does not have a universal element, see [35]. 12. Planability Considering planability of dendroids, in 1959, B. Knaster posed the following question, which is still unsolved. Problem 47. Characterize dendroids that can be embedded in the plane.

758?

In [39] T. Maćkowiak showed that there is no universal element in the class of plane smooth dendroids and in [31] L. Habiniak proved that there is no plane dendroid containing all plane smooth dendroids. Using the same definitions of Section 11 the following is still an open problem. Problem 48 ([13, p. 307]). Is the property of non-embeddability in the plane finite in the classes (e), (f ), (g), (h), (i)?

759?

It is known that the property of non-embeddability in the plane is not finite for classes (a), (c) and (j) (see [38]) and for classes (b), (d) (see [17]). Now, we move to different concepts. Lelek proved. Theorem 12.1 ([37, Theorem p. 307]). If the set E(X) of all end points of a dendroid X is not a Gδσδ -set then X is non-planable. However, the following problem is still open. Problem 49 ([19, [Problem 1]). Does there exist a dendroid X such that the set E(X) is not a Gδσδ -set?

760?

Also, T. Maćkowiak asked: Problem 50 ([39]). Is planability of dendroids (fans) an invariant property with respect to open mappings? A negative answer to Problem 50 for finite graphs is known in [53, p. 189]. Also considering continuous images of dendroids J.J. Charatonik proved in [4] that a monotone image of a planable λ-dendroid (dendroid, fan) is a planable λ-dendroid (dendroid, fan).

761–762?

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13. Shore sets A subset A of a dendroid X is said to be a shore set provided that for each ε > 0, there exists a subcontinuum B of X such that B ∩ A = ∅ and H(B, X) < ε. Answering a question of I. Puga-Espinosa, A. Illanes proved in [33] the following. Theorem 13.1. If X is a dendroid and A1 , A2 , . . . , Am are pairwise disjoint shore subcontinua of X, then A1 ∪ A2 ∪ · · · ∪ Am is a shore set. He also gave an example [33, Example 5] which shows that it is necessary to require pairwise disjointness in the previous theorem. The following natural problem is still open. 763?

Problem 51 ([33, Question 6]). Is the union of two disjoint closed shore subsets of a dendroid X also a shore set? References [1] M. Bell and S. Watson, Not all dendroids have means, Houston J. Math. 22 (1996), no. 1, 39–50. [2] J. Charatonik, J, Dendrites and montone mappings, Math. Pannon. 15 (2004), no. 1, 115– 125. [3] J. Charatonik, J and W. J. Charatonik, The property of Kelley for fans, Bull. Polish Acad. Sci. Math. 36 (1988), no. 3–4, 169–173 (1989). [4] J. J. Charatonik, A theorem on monotone mappings of planable λ-dendroids, Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys. 24 (1976), no. 3, 171–172. [5] J. J. Charatonik, Problems and remarks on contractibility of curves, General topology and its relations to modern analysis and algebra, IV (Prague, 1976), Part B, Association of Czechoslovak Mathematicians and Physicists, Prague, 1977, pp. 72–76. [6] J. J. Charatonik, Contractibility of curves, Matematiche (Catania) 46 (1991), no. 2, 559–592 (1993). [7] J. J. Charatonik, Monotone mappings of universal dendrites, Topology Appl. 38 (1991), no. 2, 163–187. [8] J. J. Charatonik, On acyclic curves. A survey of results and problems, Bol. Soc. Mat. Mexicana (3) 1 (1995), no. 1, 1–39. [9] J. J. Charatonik and W. J. Charatonik, Fans with the property of Kelley, Topology Appl. 29 (1988), no. 1, 73–78. [10] J. J. Charatonik and W. J. Charatonik, Dendrites, XXX National Congress of the Mexican Mathematical Society (Spanish) (Aguascalientes, 1997), Sociedad Matem´ atica Mexicana, M´ exico, 1998, pp. 227–253. [11] J. J. Charatonik, W. J. Charatonik, and P. Krupski, Dendrites and light open mappings, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1839–1843. [12] J. J. Charatonik, W. J. Charatonik, K. Omiljanowski, and J. R. Prajs, Hyperspace retractions for curves, Dissertationes Math. 370 (1997), 34 pp. [13] J. J. Charatonik and C. Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297–322. [14] J. J. Charatonik and Z. Grabowski, Homotopically fixed arcs and the contractibility of dendroids, Fund. Math. 100 (1978), no. 3, 229–237. [15] J. J. Charatonik and A. Illanes, Homotopy properties of curves, Comment. Math. Univ. Carolin. 39 (1998), no. 3, 573–580. [16] J. J. Charatonik and A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004), 109– 132.

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1 2 -Homogeneous

continua

Sam B. Nadler, Jr.

1. Introduction A space is homogeneous provided that for any two of its points, there is a homeomorphism of the space onto itself taking one of the points to the other point. Intuitively, spaces that are homogeneous look the same everywhere. Homogeneity is a classical topic in continuum theory. For information about homogeneous continua, I refer the reader to the article by Janusz Prajs in this book. We will discuss 12 -homogeneity, a notion that is closely related to homogeneity. We give the formal definition of 12 -homogeneity in a moment. First, we note a visible property of the closed unit n-dimensional ball B n in Euclidean n-space: For any two points in the sphere S n−1 (= ∂B n ) or in B n \ S n−1 , there is a homeomorphism of B n onto B n taking one of the points to the other, but there is no homeomorphism of B n onto B n taking a point of S n−1 to a point of B n \ S n−1 (a formal proof of the last fact uses Invariance of Domain [6, p. 95, VI9]. The abstract formulation of this property of B n is the definition of 12 -homogeneity, which we give next. Let H(X) denote the group of homeomorphisms of a space X onto itself. An orbit of X is the action of H(X) at a point x of X, meaning {h(x) : h ∈ H(X)} for a given point x ∈ X. We say X is 12 -homogeneous provided that X has exactly two orbits. More generally, for a positive integer n, X is said to be n1 -homogeneous provided that X has exactly n orbits. Thus, the 11 -homogeneous spaces are the homogeneous spaces. We give some simple examples of 12 -homogeneous continua: A figure eight (the join of two simple closed curves at a point); a θ-curve; the Hawaiian earring (a null sequence of simple closed curves joined at a point); the Sierpi´ nski universal curve [10]; the compactification of R1 with two disjoint circles as remainder for which the ray [0, ∞) continually winds in a clockwise (or counterclockwise) direction as it approaches one circle and the other ray (−∞, 0] does the same as it approaches the other circle. Regarding the last example, we note that if the rays approach the circles changing direction after each complete revolution (only), then the compactification is 13 -homogeneous. We note another situation in which two related constructions give different results: The suspension over any nonlocally connected homogeneous continuum is 12 -homogeneous, but the cone over such a continuum is not 1 2 -homogeneous when it is finite dimensional (see Theorem 4.8). Until recently, there were only two papers about 12 -homogeneity, [10, 18]. In the past two years, four more papers have been written. The four recent papers fit into three categories: 12 -homogeneous continua with cut points ([16, 17]), 12 homogeneous cones [14], and 12 -homogeneous hyperspaces [15]. We survey the 335

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main results and discuss open problems in each category separately. We provide detailed references for all results and problems that do not originate here. 2. Notation and terminology A compactum is a nonempty compact metric space. A continuum is a connected compactum. We assume that the reader is somewhat familiar with continuum theory. Most notation and terminology that we use is standard and can be found in [11, 13, 20]. However, we note the following items (other notation and terminology that is not standard is presented as it comes up): A cut point (separating point) of a connected space is a point whose removal disconnects the space. The remainder of a compactification Y of a space Z is Y \ Z. Let X be a compactification of R1 , and let R denote the open, dense copy of R1 in X. For any point r ∈ R, the closure in X of a component of R \ {r} is called an end of the compactification X. (Thus, up to homeomorphism, there are at most two ends.) The symbol ordp (X) denotes the order of the space X at p; ordp (X) ≤ ω means that p has arbitrarily small open neighborhoods whose boundaries are finite [11, p. 274]. The symbols AR and ANR stand for absolute retract and absolute neighborhood retract, respectively. A continuum Y is n-homogeneous (n a positive integer) provided that for any two n-element subsets A and B of Y , there is a homeomorphism h of Y onto Y such that h(A) = B [19]. A continuum Y is n-homogeneous at a point p ∈ Y (n a positive integer) provided that for any two n-element subsets A and B of Y such that p ∈ A ∩ B, there is a homeomorphism h of Y onto Y such that h(A) = B and h(p) = p (this notion originates in [16]). A finite graph is a 1-dimensional compact connected polyhedron. A bouquet of continua Y is a continuum X with a cut point c such that the closure of each component of X \{c} is homeomorphic to Y . The Hawaiian earring is the unique locally connected bouquet of infinitely many simple closed curves. 3.

1 2 -Homogeneous

continua with cut points

We denote the subspace of all cut points of a continuum X by Cut(X). Recall that every continuum has noncut points [13, p. 89, 6.6]; thus, when a 1 -homogeneous continuum X has cut points, the two orbits of X must be Cut(X) 2 and its complement (the set of all noncut points of X). In [16] the general stucture of 12 -homogeneous continua with cut points and the structure of their two orbits was described in detail. In addition, it was determined how the two orbits are situated in X. Nevertheless, as we will see, there are still open questions about the structure of such continua. We state the results from [16] in the four theorems that follow. The first theorem lays the foundation for the next three theorems. We note that the first theorem shows (implicitly) that if a 12 -homogeneous continuum has a cut point, then it has either uncountably many cut points or only one cut point.

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Theorem 3.1 ([16, 6.1]). Let X be a 12 -homogeneous continuum with at least one cut point. (1) If |Cut(X)| > 1, then Cut(X) is homeomorphic to R1 , Cut(X) is both open and dense in X, the orbit of all noncut points of X is the union of two disjoint, homeomorphic and homogeneous continua (possibly single points, in which case X is an arc, and the ends of the compactification X are mutually homeomorphic. (2) If Cut(X) = {c}, then the closures of the components of X \ {c} are mutually homeomorphic and are (each) 2-homogeneous at c; furthermore, if ordc (X) ≤ ω, then X is a locally connected bouquet of simple closed curves (thus, X is the Hawaiian earring when X \{c} has infinitely many components). In connection with part (2) of Theorem 3.1, we note that the closures of the components of X \ {c} need not be homogeneous: Attach two disjoint copies of a pinched 2-sphere together at the pinched points [16, 7.4]; the continuum obtained from the attachment is easily seen to be 12 -homogeneous (see Theorem 3.4). The next theorem isolates the properties in Theorem 3.1 that are relevant to the structure of Cut(X) for any 12 -homogeneous continuum. Theorem 3.2 ([16, 6.2]). If X is a 12 -homogeneous continuum, then either Cut(X) is homeomorphic to R1 and Cut(X) is both open and dense in X or Cut(X) consists of at most one point; furthermore, if Cut(X) consists of a single point c, then ordc (X) ≥ 4 and ordc (X) is even if ordc (X) is an integer. The following theorem characterizes all than one cut point:

1 2 -homogeneous

continua with more

Theorem 3.3 ([16, 6.4]). Let X be a continuum with more than one cut point. Then X is 12 -homogeneous if and only if X is an arc or X is a compactification of R1 whose remainder is the union of two disjoint, nondegenerate, homeomorphic continua and the ends of X are mutually homeomorphic and 13 -homogeneous. Our final theorem from [16] is a partial characterization of continua with more than one cut point:

1 2 -homogeneous

Theorem 3.4 ([16, 6.5]). Let X be a continuum with only one cut point c. Assume that the components of X \ {c} form a null sequence. Then X is 12 -homogeneous if and only if the closures of the components of X \ {c} are mutually homeomorphic and are (each) 2-homogeneous at c. In the following example, we show that the assumption in Theorem 3.4 that the components of X \ {c} form a null sequence is required and is restrictive. Example 3.5 ([16, 7.2 and 7.3]). Let Z = {0, 1, 12 , . . . , n1 , . . . }, let C be the Cantor set, let S 1 be the unit circle, and fix a point p ∈ S 1 . The quotient space X = (Z ×S 1 )(Z × {p}) has only one cut point c = Z ×{p} and the closures of the components of X \ {c} are mutually homeomorphic and are (each) 2-homogeneous

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at c; however, X is not 12 -homogeneous since X is locally connected at some noncut points but not at others. On the other hand, Y = (C × S 1 (C × {p}) is a 12 -homogeneous continuum with only one cut point c = C × {p0 } and, yet, the components of Y \ {c} do not form a null sequence. The continuum Y also shows that the assumption that ordc (X) ≤ ω in part (2) of Theorem 3.1 is required. Now, we come to some questions about Theorem 3.3 and Theorem 3.4. The characterization in Theorem 3.3 would be enhanced if we had a solution to the following problem (the two ways of stating the problem are equivalent by [16, 4.7]): 764?

Problem 3.6 ([16, Section 7]). Find intrinsic conditions that characterize all 13 homogeneous compactifications of [0, ∞). In other words, When is the remainder of a compactification of [0, ∞) an orbit of the compactification? It may be that any inherent characterization of 13 -homogeneous compactifications of [0, ∞) would be too technical to be useful. In fact, we do not know the answer to Problem 3.6 when the remainder of the compactification is a simple closed curve; the problem is pinpointed in the following question:

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Problem 3.7 ([16, 7.1]). Consider a compactification of the ray [0, ∞) with the circle S 1 as remainder such that every point of S 1 is a limit of points in the ray at which the ray reverses direction for at least one full revolution about S 1 . Can such a compactification be 13 -homogeneous? We ask about extending Theorem 3.4:

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Problem 3.8 ([16, Section 7]). Characterize (inherently) all 12 -homogeneous continua with only one cut point. Regarding Theorem 3.4 as well as part (2) of Theorem 3.1, we would like a solution to the following problem:

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Problem 3.9 ([16, Section 7]). Characterize (inherently) the continua that are 2-homogeneous at a point. We discuss three theorems from [17] that characterize particular continua in terms of 12 -homogeneity. The first two theorems characterize the arc. Theorem 3.10 ([17, 3.6]). The arc is only 12 -homogeneous semilocally connected continuum with more than one cut point. Theorem 3.11 ([17, 4.6]). The arc is only 12 -homogeneous hereditarily decomposable continuum whose nondegenerate proper subcontinua are arc-like. The assumption of being hereditarily decomposable in Theorem 3.11 is required: The arc of pseudo-arcs is a 12 -homogeneous arc-like continuum [17, 4.8].

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Problem 3.12 ([17, 4.9]). Is there a continuum?

1 2 -homogeneous

indecomposable arc-like

1 2 -Homogeneous

cones

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By Theorem 3.11, there is no 12 -homogeneous hereditarily decomposable circlelike continuum. The arc of pseudo-arcs with the end tranches identified to a point is an example of a 12 -homogeneous decomposable circle-like continuum [17, 4.8]. These observations lead to the following question: Problem 3.13 ([17, 4.10]). Is there a 12 -homogeneous indecomposable circle-like continuum?

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The question of determining all 12 -homogeneous arc-like continua or circle-like continua is implicit from Theorem 3.11, Problem 3.12 and Problem 3.13. Our next theorem characterizes the Hawaiian earring. Theorem 3.14 ([17, 3.12]). Let X be a 12 -homogeneous hereditarily locally connected continuum with a cut point that is not a finite graph. Then X is the Hawaiian earring. I do not know if having a cut point is required for Theorem 3.14: Problem 3.15. Is the Hawaiian earring the only 12 -homogeneous hereditarily locally connected continuum that is not a finite graph?

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Let us note a lemma that follows easily from Theorem 3.1 and Theorem 3.10: Lemma 3.16. A 12 -homogeneous finite graph with at least one cut point is either an arc or a bouquet of finitely many simple closed curves. The following variation on Theorem 3.14 is an immediate consequence of Lemma 3.16 and Theorem 3.14: Theorem 3.17. Let X be a hereditarily locally connected continuum with a cut point. Then X is 12 -homogeneous if and only if X is an arc or a bouquet of simple closed curves (that is, a finite bouquet or the Hawaiian earring). Lemma 3.16 raises a question about finite graphs. Patkowska [18, p. 25, Theorem 1] claims “Moreover, we find a full classification of all 12 -homogeneous polyhedra by means of homogeneous multigraphs.” However, the meaning of the claim (and its verification) does not seem to be in [18], even for the case of finite graphs. Lemma 3.16 takes care of the case of finite graphs with a cut point; however, we do not know about cyclic finite graphs: Problem 3.18. What are all the 12 -homogeneous finite graphs that have no cut point? 4.

1 2 -Homogeneous

cones

We denote the cone over a compactum X by Cone(X) and its vertex by vX . The question of when the cone over a continuum is 12 -homogeneous was investigated for the first time in [15]. So far, there are no other papers about this topic. The main results in [15] fall into three categories: 1-dimensional continua, an ANR theorem for finite-dimensional compacta, and continua with conditions

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weaker than being atriodic in some nonempty open set. We discuss many of the results and open problems from [15]. Note that Cone(B n ) and Cone(S n ) are (n+1)-cells and, hence, are 12 -homogeneous. One of the main results from [15] is that B 1 and S 1 are the only 1dimensional continua whose cones are 12 -homogeneous: Theorem 4.1 ([15, 6.1]). Let X be a 1-dimensional continuum. Then Cone(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. The natural analogue for all finite-dimensional continua of Theorem 4.1 is false: Example 4.2 ([15, 1.1]). For each integer n ≥ 4, let X = Cone(S n−1 /A), where A is an arc in the (n−1)-sphere S n−1 such that the fundamental group of S n−1 \ A is nontrivial. Then X is an n-dimensional AR that is not a manifold and, yet, Cone(X) is 12 -homogeneous since Cone(X) an (n+1)-cell [2, p. 26, 4.4]. Theorem 4.1 and Example 4.2 lead us to a question for dimensions 2 and 3 as well as a question for any finite dimension: 772?

Problem 4.3 ([15, 1.2]). If X is a continuum, even a Peano continuum, of dimension n = 2 or 3 such that Cone(X) is 12 -homogeneous, must X be an n-cell or an n-sphere?

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Problem 4.4 ([15, 1.3]). If the cone over a finite-dimensional continuum, even a Peano continuum, is 12 -homogeneous, then is the cone an n-cell? We note several corollaries to Theorem 4.1. (It is not obvious why Corollary 4.5 is a consequence of Theorem 4.1; to see why it is uses some technical lemmas that we do not include here.) Corollary 4.5 ([15, 6.2]). Let X be a nondegenerate continuum that contains only finitely many simple closed curves. Then Cone(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. Corollary 4.6 ([15, 6.3]). If X is a nondegenerate tree-like continuum, then Cone(X) is 12 -homogeneous if and only if X is an arc. Corollary 4.7 ([15, 6.7]). The only circle-like continuum whose cone is 12 -homogeneous is a simple closed curve. Next, we turn our attention to the class of finite-dimensional compacta whose cones are 12 -homogeneous. We begin with an ANR theorem and a corollary for all finite-dimensional compacta; we show that the theorem and the corollary do not extend to infinite-dimensional compacta. Theorem 4.8 ([15, 3.5]). Let X be a finite-dimensional compactum. If Cone(X) is 12 -homogeneous, then X is an ANR. Corollary 4.9 ([15, 3.7]). If X is a finite-dimensional compactum such that Cone(X) is 12 -homogeneous, then Cone(X) is an AR.

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Theorem 4.8 and Corollary 4.9 do not extend to infinite dimensions even when X is locally connected. We will give an example that is based on the following result (which we state slightly differently than in [15]): Theorem 4.10 ([15, 3.8]). If Y is a homogeneous compactum and Q is the Hilbert cube, then Cone(Y × Q) is either homogeneous or 12 -homogeneous. Example 4.11 ([15, 3.10]). Let X = M × Q, where M is the 1-dimensional Menger universal curve and Q is the Hilbert cube. Then X is locally connected and Cone(X) is 12 -homogeneous (by Theorem 4.10), but X and Cone(X) are not ANRs. Furthermore, Theorem 4.8 and Corollary 4.9 fail badly in infinite dimensions in that X (hence, Cone(X)) need not even be locally connected: Let X = Y × Q, where Y is a nonlocally connected homogeneous continuum Y (e.g., the dyadic solenoid or the pseudo-arc [3]); then Cone(X) is 12 -homogeneous (by Theorem 4.10), but X and Cone(X) are not locally connected. We note another corollary to Theorem 4.8 and a problem concerning the corollary. Corollary 4.12 ([15, 3.6]). If X is a finite-dimensional contractible continuum such that Cone(X) is 12 -homogeneous, then X is an AR. Unlike Theorem 4.8 and Corollary 4.9, we do not know if Corollary 4.12 extends to infinite dimensions: Problem 4.13 ([15, 3.11]). If X is a contractible continuum such that Cone(X) is 12 -homogeneous, then is X an AR? What about with the additional assumption that X is locally connected? Finally, we discuss results for continua that satisfy conditions that are weaker than being atriodic. For the first result, we note names for two special continua: (1) the hairy point is the union of a null sequence of countably infinitely many arcs all emanating from the same point and otherwise disjoint from one another; (2) the null comb is the continuum homeomorphic to the union of the line segments in the plane from (0, 0) to (1, 0) and from ( n1 , 0) to ( n1 , n1 ) for each n = 1, 2, . . . . Theorem 4.14 ([15, 6.4]). Let X be a nondegenerate continuum with a nonempty open set U such that U does not contain a hairy point or a null comb. Then Cone(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. Corollary 4.15 ([15, 6.5]). Let X be a nondegenerate continuum that contains no simple triod in some nonempty open set U . Then Cone(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. Corollary 4.16 ([15, 6.6]). Let X be a nondegenerate continuum with a nonempty open set U such that every nondegenerate subcontinuum of U is arc-like. Then Cone(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. We complete this section by stating two problems that are natural from what we have discussed (the problems are not explicitly stated elsewhere).

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Problem 4.17. Find more classes of continua whose cones are 12 -homogeneous.

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Problem 4.18. Find classes of continua whose suspensions are 12 -homogeneous. 5.

1 2 -Homogeneous

hyperspaces

For a continuum X with metric d, the hyperspace C(X) is the space of all subcontinua of X with the Hausdorff metric ([9] or [12]). It has been known for some time when C(X) is homogeneous, namely, if and only if X is a Peano continuum in which every arc is nowhere dense or, equivalently, C(X) is the Hilbert cube. (This was proved in [12, p. 564, 17.2] using [5, p. 22, 4.1].) The next logical step from the point of view of homogeneity-type properties is to inquire into when C(X) is 12 -homogeneous. This was investigated for the first time in [14]. One of our principal tools used in [14] is the theory of layers (or tranches) of irreducible hereditarily decomposable continua [11, pp. 190–219]. Two simple continua for which C(X) is 12 -homogeneous are the arc and the simple closed curve; in both cases, C(X) is a 2-cell [9, pp. 33–35]. The two main results in [14], which we state next, suggest that there are very few continua X for which C(X) is 12 -homogeneous and, in fact, that the arc and the simple closed curve may be the only ones. Theorem 5.1 ([14, 3.1]). If X is a locally connected continuum, then C(X) is 1 2 -homogeneous if and only if X is an arc or a simple closed curve. Theorem 5.2 ([14, 5.1]). Let X be a nondegenerate atriodic continuum. Then C(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. Corollary 5.3 ([14, 5.2]). Let X be a continuum such that each nondegenerate proper subcontinuum of X is arc-like. Then C(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. Corollary 5.3 shows that when X is arc-like (circle-like, atriodic tree-like), then C(X) is 12 -homogeneous if and only if X is an arc (a simple closed curve, an arc, respectively) [14, 5.3, 5.4, 5.6]. Corollary 5.4 ([14, 5.7]). Let X be a continuum such that dim C(X) = 2. Then C(X) is 12 -homogeneous if and only if X is an arc or a simple closed curve. 778–779?

Problem 5.5 ([15, Section 1]). If X is a continuum such that C(X) is 12 homogeneous, then is X an arc or a simple closed curve? What about when C(X) is finite dimensional? We note two results that give information about the second part of Problem 5.5 (two other such results are [14, 3.9 and 3.12]). Theorem 5.6 ([14, 3.10]). Let X be a decomposable continuum such that dim C(X) < ∞. If C(X) is 12 -homogeneous, then X is hereditarily decomposable. Theorem 5.7 ([14, 3.11]). Let X be a nonlocally connected continuum such that dim C(X) < ∞ and C(X) is 12 -homogeneous. Then every nondegenerate proper subcontinuum of X is decomposable.

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A number of questions were asked at the end of [14]. The purpose of some of the questions was to indicate directions that might lead to solutions or partial solutions to Problem 5.5. We summarize a few such questions: Problem 5.8 ([14]). Let X be a continuum such that C(X) is 12 -homogeneous. Is dim C(X) < ∞ and, in fact, is dim C(X) = 2 [14, 6.2]? Is X decomposable [14, 6.4]? Is dim X = 1 and, in fact, must X be hereditarily decomposable [14, 6.5]? Must X be hereditarily decomposable when dim C(X) < ∞ [14, 6.5]?

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(Note: In [14], the third part of 6.5 says, “Is dim C(X) < ∞?”, which was already asked in [14, 6.2]; the way the third part of [14, 6.5] is stated in Problem 5.8 is what was meant.) Regarding Problem 5.8, X can not be hereditarily indecomposable [14, 3.3]. However, we do not know if X can contain a nondegenerate hereditarily indecomposable continuum—if it does not, then dim X = 1 [4, p. 270, Theorem 5]. In investigating when C(X) is 12 -homogeneous, it is important to have conditions under which various elements of C(X) belong or do not belong to the manifold interior of a 2-cell. Acosta showed that if X is an atriodic continuum, then no singleton, {x}, belongs to the manifold interior of any 2-cell in C(X) (weakened form of [1, p. 40, Theorem 3]). Acosta’s result was important for the proof of Theorem 5.2; in view of that, the following question was asked in [14]: Problem 5.9 ([14, 6.6]). What conditions on continua X (other than being atriodic) or on points p ∈ X are necessary and/or sufficient for {p} not to belong to the manifold interior of a 2-cell in C(X)?

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A point p of a finite graph X is as in Problem 5.9 if and only if ordp (X) ≤ 2; however, for the point p = (0, 0) in the null comb X (defined preceding Theorem 4.14), ordp (X) = 1 and, yet, {p} belongs to an n-cell in C(X) for every n by [9, p. 40, 6.4]. As noted in [14], Problem 5.9 for n-cells in C(X) is open as well. We also note the following question about 2-cells in C(X): Problem 5.10 ([14, 6.7]). Is there a continuum X such that dim C(X) < ∞ and, for every x ∈ X, {x} is a point of the manifold interior of a 2-cell in C(X)?

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There is no reason to restrict the study of 12 -homogeneous hyperspaces to the hyperspace C(X). Several special hyperspaces other than C(X) are of general interest—the hyperspace 2X of all nonempty compact subsets of a continuum X (with the Hausdorff metric), the n-fold hyperspace Cn (X) of all elements of 2X with at most n components, and the n-fold symmetric product Fn (X) of all elements of 2X with at most n points. Our final question concerns these hyperspaces. Problem 5.11 ([14, 6.8]). For what continua X are the hyperspaces 2X , Cn (X) 1 or Fn (X) 12 -homogeneous (n > 1 for the case of Fn (X))? What about m homogeneity for any integer m > 1? is

The case of 12 -homogeneity for 1 2 -homogeneous since C2 ([0, 1])

C2 (X) seems especially interesting: C2 ([0, 1]) is a 4-cell ([7, p. 349, Lemma 2.2], due to

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R.M. Schori); however, C2 (S 1 ) is not 12 -homogeneous since C2 (S 1 ) is the cone over a solid torus [8]. This (naively) suggests that C2 (X) may only be 12 -homogeneous when X is an arc. In fact, this is true when X is locally connected (a proof is in the comments following Question 6.8 of [14]). References [1] G. Acosta, Continua with unique hyperspace, Continuum theory (Denton, TX, 1999), Lecture Notes in Pure and Appl. Math., vol. 230, Marcel Dekker Inc., New York, 2002, pp. 33–49. [2] F. D. Ancel and S. B. Nadler, Jr., Cones that are cells, and an application to hyperspaces, Topology Appl. 98 (1999), 19–33. [3] R H Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. [4] R H Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267–273. [5] D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), no. 1, 19–38. [6] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, NJ, 1941. [7] A. Illanes, The hyperspace C2 (X) for a finite graph X is unique, Glas. Mat. Ser. III 37(57) (2002), no. 2, 347–363. [8] A. Illanes, A model for the hyperspace C2 (S 1 ), Questions Answers Gen. Topology 22 (2004), no. 2, 117–130. [9] A. Illanes and S. B. Nadler, Jr., Hyperspaces, Marcel Dekker Inc., New York, 1999. [10] J. Krasinkiewicz, On homeomorphisms of the Sierpi´ nski curve, Prace Mat. 12 (1969), 255– 257. [11] K. Kuratowski, Topology. Volume II, Academic Press, New York, 1968. [12] S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker Inc., New York, 1978. [13] S. B. Nadler, Jr., Continuum theory, Marcel Dekker Inc., New York, 1992. [14] S. B. Nadler, Jr. and P. Pellicer-Covarrubias, Hyperspaces with exactly two orbits, Glasnik Mat. 41 (2006), 141–157. [15] S. B. Nadler, Jr. and P. Pellicer-Covarrubias, Cones that are 12 -homogeneous, Houston J. Math., To appear. [16] S. B. Nadler, Jr., P. Pellicer-Covarrubias, and I. Puga, 12 -homogeneous continua with cut points, Topology Appl., To appear. [17] V. Neumann-Lara, P. Pellicer-Covarrubias, and I. Puga, On 12 -homogeneous continua, Topology Appl. 153 (2006), no. 14, 2518–2527. [18] H. Patkowska, On 12 -homogeneous ANR-spaces, Fund. Math. 132 (1989), no. 1, 25–58. [19] G. S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 393–400. [20] G. T. Whyburn, Analytic topology, American Mathematical Society, Providence, RI, 1942.


Thirty open problems in the theory of homogeneous continua Janusz R. Prajs Broadly understood symmetry is an archetypical quality abundant both in nature and human creativity. In particular, its presence in mathematics is overwhelming. Functions, formulas and spaces with special symmetric properties, as a rule, tend to be more important and have more applications than others. In geometry, symmetry manifests through invariance with respect to certain isometric transformations. Since the concept of an isometry is not topological, one can ask what topological properties could possibly represent symmetry in this broad meaning. Which topological spaces would have strong symmetric properties? We propose the following answer: the richer the group of self-homeomorphisms of a topological space, the more symmetric the space. This answer, which naturally corresponds to geometric symmetry, leads us to the concept of topological homogeneity introduced by Sierpi´ nski [25, p. 16]. A topological space X is homogeneous provided for each x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. This definition identifies a fundamental class of spaces with rich groups of self-homeomorphisms. The systematic study of homogeneous spaces began with the question of Knaster and Kuratowski [6] whether the simple close curve is the only nondegenerate, homogeneous plane continuum. Since then, classifying homogeneous continua became a classic topic, which now is an important area in continuum theory. The restriction to the study of homogeneous continua is reasonable indeed. First, the class of all homogeneous spaces is so vast, that one cannot expect many important results about that class as whole. Thus the class of compact, metrizable spaces, which have the most common applications, is a natural choice. As it was shown by Michael Mislove and James Rogers [13, 14], each compact, metrizable homogeneous space is a product of a finite set or the Cantor set, and a homogeneous continuum, that is, a homogeneous compact, connected, metric space. This makes investigating homogeneous continua particularly important. Let us notice that the class of homogeneous continua is a natural generalization of the two following important classes of spaces, both in the focus of classic, mainstream study in topology and mathematics: (1) closed, connected manifolds, and (2) compact, connected topological groups (including Lie groups). The significance of these classes provides further motivation for the study of homogeneous continua.

The author was supported in part by the NSF grant DMS-0405374. The author was supported in part by assigned time from the Chair of the Department of Mathematics and Statistics, Dr. Doraiswamy Ramachandran, at California State University, Sacramento 345

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§37. Prajs, Thirty open problems in the theory of homogeneous continua

Despite excellent motivation and persistent effort since early 1920s, progress in understanding homogeneous continua and finding new examples has been slow, though rewarded with occasional unexpected turns and spectacular breakthroughs. The purpose of this article is to contribute to this effort by offering questions and possible directions for future research. We do not focus our attention, however, on the most classic questions such as the ones about homogeneous plane continua, (hereditarily) indecomposable homogeneous continua, or hereditarily decomposable homogeneous continua. Excellent references to these problems can be found in [9, 12, 23]. The problems collected in this paper are divided into two parts. In Section 2 we present miscellaneous problems, some of which already have been published, and some are new. In the author’s view, these questions may have potential to become a part of the mainstream study of homogeneous continua in the future. The remaining part of the paper is devoted to a new line of study of homogeneous continua, initiated in [20, 21] and based on the duality of filament and ample subcontinua. As it is shown in [22], this new research is related to the past applications of aposyndesis to homogeneous continua. The idea is to identify major archetypical classes of homogeneous continua related to the structure of their filament subcontinua, and investigate properties of these classes. After presenting definitions and summary of basic facts in Section 3, we propose and discuss questions related to this new approach in Section 4. The author acknowledges collaboration with Keith Whittington on filament and ample subcontinua of homogeneous continua. In fact, a substantial part of the questions from Section 4 emerged from this collaboration.

1. Preliminaries A continuum is a compact, connected, nonempty metric space. Continua of dimension 1 are called curves. If X is a continuum, C(X) will denote the hyperspaces consisting of all subcontinua of X under the Hausdorff metric. The definition of a homogeneous space is given in the introduction. A space X is 2homogeneous if for every x1 , x2 , y1 , y2 ∈ X with x1 6= x2 and y1 6= y2 there exists a homeomorphism h : X → X such that h({x1 , x2 }) = {y1 , y2 }. Though we do not explicitly use the Effros Theorem in this paper, it is a fundamental tool applied in the proofs of many cited results, and can be very helpful when attacking problems involving homogeneous continua. Therefore we recall it here. If X is a homogeneous continuum, then for every positive ε, there is a number δ, called an Effros number for ε, such that for each pair of points with d(x, y) < δ, there is some homeomorphism f : X → X that carries x to y and such that d(z, f (z)) < ε for each z ∈ X. This is called the Effros Theorem. It follows from the more general statement that for each x ∈ X, the evaluation map, g 7→ gx, from the homeomorphism group onto X is open. The latter follows from [5, Theorem 2]. (See also [26, Theorem 3.1].)

Fourteen miscellaneous problems

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2. Fourteen miscellaneous problems By Mazurkiewicz’s theorem [11] the simple closed curve is the only locally connected, nondegenerate homogeneous continuum in the plane. An analogous result in 3-space is yet to be found. Note that, by Anderson’s result [1], 1-dimensional locally connected continua are precisely the simple closed curve and Menger curve. In the first question the Pontryagin sphere appears. The Pontryagin sphere has several equivalent definitions. For instance, let S be the Sierpi´ nski universal plane curve, also known as Sierpi´ nski’s carpet, in its standard geometric construction in the unit square [0, 1] × [0, 1]. The quotient space obtained from S by identifying each pair of points a, b such that a and b are in the boundary of the same complementary domain of S in the plane, and a and b have at least one coordinate the same, is a Pontryagin sphere. Another way to define the Pontryagin sphere is to take two Pontryagin disks defined in [15, pp. 608–609] and glue them together along their combinatorial boundary. It is known that the Pontryagin sphere is homogeneous. Question 1. If X is a homogeneous, locally connected, 2-dimensional continuum in 3-space, is X either a 2-manifold, or a Pontryagin sphere?

788?

Question 2. Is every nondegenerate, simply connected homogeneous continuum in 3-space homeomorphic to 2-sphere S2 ?

789?

The four next questions refer to the important class of path-connected homogeneous continua, and they are essential in the non-locally connected case. Krystyna Kuperberg asked [7, Problem 2, p. 630] whether each path-connected homogeneous continuum is locally connected. This question was answered in the negative in [17]. The following related question seems to provide a similar type of challenge. Question 3. If X is a simply connected homogeneous continuum, is X locally connected?

790?

The following question was explored in the past by David Bellamy, who obtained a strong partial result [2]. Question 4. If X is a path-connected homogeneous continuum, is X uniformly path-connected?

791?

Since all uniformly path-connected continua are weakly chainable, a positive answer to the previous question would imply one to the next question. Question 5. If X is a path-connected homogeneous continuum, is X weakly chainable? The path-connected example P from [17] has a natural projection onto the Menger curve such that P has a unique path lifting property with respect to this projection. It is not known whether each homogeneous path-connected continuum admits such a map. We ask the following.

792?

348 793?


Question 6. Let X be a homogeneous path-connected (1-dimensional) continuum. Does X admit an open surjective map f : X → Y onto a locally connected continuum Y such that X has the unique path lifting property with respect to f ? The next three questions are related to each other. They ask about the existence of certain inverse limit representations for some homogeneous continua. To formulate the first of these problems, which originally appeared in [16], we need some definitions. A surjective map f : X → Y is called confluent if for every continuum K in Y and every p ∈ f −1 (K) there exists a continuum C ⊂ X such that p ∈ C and f (C) = K. A continuum X is confluently graph-like provided for every ε > 0 there is a confluent map of X to a graph with point inverses having diameters less than ε. A continuum is called confluently graph-representable if it can be represented as the inverse limit of graphs with confluent bonding maps. By one of the main results of [16] the property “confluently graph-like” is equivalent to “confluently graph-representable” for continua.

794?

Question 7. If X is a homogeneous curve that contains an arc, is X confluently graph-like?

795?

Question 8 (J.H. Case [4]). If X is a homogeneous curve that contains an arc, can X be represented as inverse limit of either simple closed curves or topological Menger curves with covering bonding maps?

796?

Question 9. If X is a homogeneous continuum such that each point of X has a neighborhood whose components are n-manifolds (Menger manifolds, Hilbert cube manifolds), is X the inverse limit of n-manifolds (Menger manifolds, Hilbert cube manifolds) with covering bonding maps? Known examples suggest that the three following questions may admit positive answers. A counterexample would provide an even more spectacular result.

797?

Question 10. Does every nondegenerate (1-dimensional) homogeneous continuum have a nondegenerate weakly chainable subcontinuum?

798?

Question 11. Does every nondegenerate homogeneous continuum contain either an arc or a nondegenerate (hereditarily) indecomposable subcontinuum?

799?

Question 12. Does every homogeneous curve contain either an arc or a proper, nondegenerate terminal subcontinuum? The remaining two problems in this section are new. The next one, interesting by its own right, appears in connection to the study of filament sets, and is related to Problems 22 and 23 from Section 4.

800?

Question 13. If a homogeneous continuum X is a finite (equivalently, countable) union of its indecomposable subcontinua, is X indecomposable? We say the group of self-homeomorphisms H(X) of a space X respects a partition G of X if h(G) ∈ G for every h ∈ H(X) and G ∈ G. Given a subcontinuum K of a space X, let HK = {h(K) : h ∈ H(X)}. For every x, y ∈ X we write

Filament sets: definitions and basic properties

349

x ∼K y provided that x = y or there are continua K1 , . . . , Kn ∈ HK such that K1 ∪ · · · ∪ Kn is connected and x, y ∈ K1 ∪ · · · ∪ Kn . Note that ∼K is an equivalence. The equivalence classes of ∼K are called K-components. The space X is K-connected if X is the only K-component in X. It is an immediate observation the partition into K-components is respected by self-homeomorphisms of X. If K1 , K2 are two subcontinua of a continuum X, we write K1 ' K2 provided the K1 -components and K2 -components are identical. Note that ' is an equivalence in C(X). We have the trivial structure of {p}-components generated by singletons {p}, which is the trivial decomposition into singletons, and which we usually ignore. For every homogeneous space X we assign the cardinality κ(X) of the collection of the equivalence classes of ' represented by nondegenerate subcontinua of X. Thus κ(X) = 0 when X is a singleton. It can easily be observed that κ(S1 ) = 1 for the unit circle S1 . As an exercise please note that κ(X) ≥ 2 if X is indecomposable, and κ(X) ≥ 3 if X is the circle of pseudo-arcs. (In fact κ(X) = 3 in the latter case.) The following proposition can easily be shown. Proposition 2.1. If X is a nondegenerate 2-homogeneous continuum, then κ(X) = 1. In particular, if X is either a manifold, the Menger curve, or the Hilbert cube, then κ(X) = 1. Using the Homeomorphism Extension Theorem [10], one can show that for the Menger curve M we have κ(M × M) = 1, even though M × M is not 2homogeneous [8]. Question 14. Let X be a homogeneous continuum with κ(X) = 1. Must X be path-connected? Must X be locally-connected? 3. Filament sets: definitions and basic properties In this section we provide basic concepts and facts involved in a new line of study of homogeneous continua, initiated in [20, 21] and based on the duality of filament and ample subcontinua. We begin with the following definitions of certain subsets of a continuum X, which are crucial in the remaining part of the paper. With an exception of (iv), they were introduced in [20]. (i) A subcontinuum F of X is called filament if there exists a neighborhood N of F such that the component of N containing F has empty interior. (ii) A set Y ⊂ X is called filament if every subcontinuum of Y is filament in X. (iii) A set Z ⊂ X is called co-filament if X \ Z is a filament set in X. (iv) A subcontinuum G of X is called almost filament if G is the limit, in the sense of the Hausdorff distance, of filament continua in X. (v) A subcontinuum A of X is called ample if every neighborhood N of A contains a continuum B such that A ⊂ int(B) ⊂ B ⊂ N . The three following propositions summarize the most fundamental properties [20] of the introduced concepts. Part (b) of Proposition 3.1 was originally proved in [27].

801–802?

350


Proposition 3.1. Let X be a homogeneous continuum. (a) A subcontinuum K of X is ample if and only if K is not filament. (b) The set of all pairs (x, y) ∈ X × X such that only ample subcontinua of X can contain both x and y is a dense Gδ subset of X × X. (c) Each ample subcontinuum of X contains a minimal ample subcontinuum. (d) Each closed filament set in X has a filament neighborhood in X. (e) The collection of filament subcontinua of X is an open, connected subset of C(X). (f) The subspace of C(X) of ample subcontinua of X is a compact absolute retract. Proposition 3.2. For every continuum X the following conditions are equivalent: (a) X is indecomposable. (b) X is the only ample subcontinuum of X. (c) Every nonempty subset of X is co-filament. Proposition 3.3. For every homogeneous continuum X the following conditions are equivalent: (a) X is locally connected. (b) Every subcontinuum of X is ample. (c) X is the only closed, co-filament subset of X. Next we define filament composants in continua, which generalize the concept of composants of indecomposable continua (see Proposition 3.4 below). (vi) Given a point p ∈ X, the union of the filament continua in X containing p is called the filament composant of X determined by p, and denoted by Fcs(p). Please note that the filament composants of a decomposable continuum are different from the (usual) composants. We recall the following fundamental properties of filament composants (see [20, Proposition 1.8]). Proposition 3.4. Let X be a continuum and p ∈ X. If Fcs(p) is nonempty, it is a countable union of filament continua, each containing p. Thus each filament composant is a first-category Fσ subset of X. If X is indecomposable, the composants and filament composants of X are identical. Employing the concept of a filament continuum, we define some classes of continua. (vii) A continuum X is filament additive provided for each two filament subcontinua F1 and F2 with nonempty intersection, the union F1 ∪ F2 is filament. (viii) A continuum X is called filament connected if for each two points p, q ∈ X there are filament continua F1 , . . . , Fn in X such that p, q ∈ F1 ∪ · · · ∪ Fn and the union F1 ∪ · · · ∪ Fn is connected.

Filament sets: definitions and basic properties

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Most known homogeneous curves are filament additive. The first non-filament additive homogeneous curve was defined in [17]. In higher dimensions, each product of at least two homogeneous, non-locally connected continua is non-filament additive [21]. Filament additive continua and filament connected continua are disjoint classes of spaces. (ix) A continuum X is called filamentable if either X is a singleton, or X has a filament subcontinuum whose complement is filament. The diagram in Figure 1 below represents a classification scheme of homogeneous continua introduced in [19]. It is based on the concept of a co-filament continuum, that is, a co-filament, compact, connected set. Homogeneous continua form a spectrum having at its ends Class I with the richest collection of co-filament subcontinua, and Class IV with the smallest one. The following conditions define the corresponding classes: (I) Every subcontinuum is co-filament; (II) Contains non-co-filament subcontinua, and also subcontinua that simultaneously are co-filament and filament; (III) All co-filament subcontinua are ample and some of them are proper ; and (IV) The whole space is the only co-filament subcontinuum. Homogeneous Continua Filamentable Nonfilamentable Class I Class II Class III Class IV Ia: singleton, IIa: filamentable, IIIa: nonfilaIVa: locally indecomposable, decomposable, mentable, with connected, filamentable, aposyndetic proper co-filament nondegenerate, aposyndetic subcontinua, nonfilamentable, aposyndetic aposyndetic Ib: indecomposable, IIb: filamentable, IIIb: nonfilaIVb: no proper nondegnerarate, decomposable, mentable, with co-filament filamentable, nonaposyndetic proper co-filament subcontinua, nonaposyndetic subcontinua, nonfilamentable, nonaposyndetic nonaposyndetic Figure 1. Classification of homogeneous continua The eight classes indicated in the diagram (Figure 1) are mutually disjoint and each of them is nonempty. If a continuum belongs to a class labeled with b, its aposyndetic decomposition quotient space is in the corresponding class labeled with a. By a recent result of Rogers [24], and by Anderson’s characterization of locally connected homogeneous curves [1], all members of Class IVb have their aposyndetic quotient spaces homeomorphic to either the circle S1 , or the Menger curve M. Below, we list at least one example of spaces belonging to each class. Note that the selected examples have dimension less than or equal to 1.


352

(Ia) (Ib) (IIa) (IIb)

A singleton. The pseudo-arc and solenoids. The Case continuum. The continuous curve of pseudo-arcs with the Case continuum as the quotient space. (IIIa) The path-connected continuum P from [17]. (IIIb) The continuous curve of pseudo-arcs with P as the quotient space. (IVa) The circle S1 and Menger curve M. (IVb) The continuous curves of pseudo-arcs with S1 and M as the quotient spaces. Finally, note that the extreme classes, Classes I and IV, are exclusively composed of filament additive continua. Thus the property “filament connected” can only occur in Classes II and III. Each of the Classes IIa, IIb, IIIa and IIIb has both filament additive and non-filament additive members. 4. Filament sets: sixteen questions In the previous section we presented the most fundamental concepts and facts related to the new line research, in the area of homogeneous continua, introduced in [20, 21] and continued in [18, 19, 22]. In this section we collect problems that are related to this new research. We begin with a question posed in [21]. 803?

Question 15. Is every homogeneous continuum either filament additive or filament connected? This intriguing problem has a positive solution in Classes I, II and IV. Obviously, in the filament additive part of Class III this question is also answered in the affirmative. It is interesting that Class III is the only one of the four, where some other problems remain unsolved. For instance, a possible counterexample to a classic question by J´ ozef Krasinkiewicz and Piotr Minc whether a nondegenerate, hereditarily decomposable, homogeneous continuum must be a simple closed curve, would have to be in the non-filament additive part of Class III [19, 21]. It is not accidental that the path-connected continuum from [17] is again in that part of Class III because every non-locally connected, path-connected homogeneous continuum is in there [19]. Class III and, in particular, its non-filament additive part remain mystery areas, which deserve special attention in the future. The four following problems seem to be essential to understand the filament structure of homogeneous continua.

804?

Question 16. If K is a subcontinuum of an almost filament continuum L in a homogeneous continuum X, is K almost filament?

805?

Question 17. If X is a homogeneous continuum with dense filament composants, is X almost filament?

806?

Question 18. If X is a homogeneous continuum and x ∈ X, is the filament composant Fcs(x) a first-category subset of the closure cl (Fcs(x))?

Filament sets: sixteen questions

353

Question 19. If X is a homogeneous, non-locally connected continuum, does there exist a nondegenerate subcontinuum K of X such that for every filament subcontinuum F of X intersecting K the union K ∪ F is filament?

807?

The two next questions are about products. If K is a subcontinuum of the product X × Y of continua X and Y , and at least one of the two projections of K is a filament subcontinuum of the corresponding space, then K is filament in X × Y [21]. The converse is not necessarily true. Indeed, David Bellamy and Janusz Lysko observed in [3] that if X is a non-circle solenoid, then the diagonal of the product X × X is filament even though both projections are ample in the corresponding spaces. In view of these facts the following question is of interest. Question 20. Let X and Y be homogeneous continua, πX : X × Y → X and πY : X × Y → Y the projections, and F a filament subcontinuum of the product X × Y . Is either πX (F ) almost filament in X, or πY (F ) almost filament in Y ?

808?

A number of rules are known that indicate where the product of given two homogeneous continua X and Y may belong, in the diagram from the previous section. For example (IVa) × (IVa) ⊂ (IVa), which is well known, and it means that if X, Y ∈ Class IVa, then X × Y ∈ Class IVa. In [19] it is shown that (IVa) × (IVb) ⊂ (IIIa). It is also observed that if X is filamentable, then so is X × Y . Thus, for instance, (IIa) × (IIIb) ⊂ (IIa) and (IIb) × (IVb) ⊂ (IIa), etc. The following is an interesting open question in this area. Question 21. If X and Y are nonfilamentable homogeneous continua, is the product X × Y nonfilamentable?

809?

The next four questions are interconnected. It can be observed that in a homogeneous continuum a minimal ample subcontinuum with nonempty interior would have to be indecomposable. The existence of such subcontinuum would imply that the space is the finite union of indecomposable subcontinua. Thus the four following questions are also related to Question 13. Question 22. If X is a homogeneous continuum such that a minimal ample subcontinuum of X has nonempty interior, must X be indecomposable?

810?

It can be shown that if a homogeneous continuum has a finite co-filament subset, then each minimal ample subcontinuum has nonempty interior. Therefore, the next question would be answered in the affirmative if the previous one was. Question 23. If X is a homogeneous continuum having a finite co-filament set C, must X be indecomposable? What if C has at most two elements?

811?

In the next two questions we focus on some converse directions to the ones of Questions 22 and 23, respectively. Question 24. If Y is an indecomposable subcontinuum, with nonempty interior, of a homogeneous continuum X, must Y be a minimal ample subcontinuum of X?

812?

354 813?


Question 25. Let X be a homogeneous continuum such that every closed cofilament set in X is infinite. Does every minimal ample subcontinuum of X have empty interior in X? In a filament additive continuum all minimal ample subcontinua are indecomposable [21]. It is not known whether the converse is true.

814?

Question 26. Let X be a homogeneous continuum such that every minimal ample subcontinuum of X is indecomposable. Must X be filament additive? In [20] it has been proved that for a homogeneous continuum X the collection A(X) of ample subcontinua of X, as a subspace of C(X), is an AR. It is interesting to ask the following.

815?

Question 27. Is the collection A(X) of ample subcontinua of a homogeneous continuum X a deformation retract of C(X)? The only known examples of homogeneous continua having the collection of minimal ample subcontinua closed belong to Class IV. Therefore, the following question naturally appears.

816–818?

Question 28. Let X be a homogeneous continuum, A0 (X) be the collection of minimal ample subcontinua of X, and assume A0 (X) is a closed subset of C(X). Is A0 (X) a partition of X? Is A0 (X) the Jones aposyndetic decomposition of X? Does X belong to Class IV? Our knowledge about the important class of homogeneous path-connected continua, especially in the non-locally connected case, is still very limited. It may be worth to explore the direction of the following question.

819?

Question 29. Let X be a homogeneous continuum having an ample, locally connected (path-connected) subcontinuum. Is X path-connected? The last question of the paper is related to Question 9 from Section 2. It employs the concept of micro-local connectedness. A continuum X is micro-locally connected at p provided there exists an open neighborhood U of p such that the component of U containing p is locally connected at p. The micro-local connectedness at p implies that X is micro-locally connected (everywhere) whenever X is homogeneous. Note that X from Question 9 is micro-locally connected. In the following question, the micro-local connectivity of the space implies the filament local product structure [18], i.e., points have neighborhoods homeomorphic to the product K × C, where K is a continuum and C is the Cantor set. Moreover, the micro-local connectivity of the space also implies that K can be locally connected. In Question 9, additionally, K can be an n-cell (a Menger continuum, the Hilbert cube). For instance, solenoids and the Case continuum are spaces for which the hypotheses of Questions 9 and 30 hold.

820?

Question 30. If X is a micro-locally connected homogeneous continuum, is X the inverse limit of locally connected continua with covering bonding maps?

References

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References [1] R. D. Anderson, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1–16. [2] D. P. Bellamy, Short paths in homogeneous continua, Topology Appl. 26 (1987), no. 3, 287–291. [3] D. P. Bellamy and J. M. Lysko, Connected open sets in products of indecomposable continua, 2005, Preprint. [4] J. H. Case, Another 1-dimensional homogeneous continuum which contains an arc, Pacific J. Math. 11 (1961), 455–469. [5] E. G. Effros, Transformation groups and C ∗ -algebras, Ann. of Math. (2) 81 (1965), 38–55. [6] B. Knaster and C. Kuratowski, Probl´ eme 2, Fund. Math. 1 (1920), 223. [7] K. Kuperberg, A locally connected microhomogeneous nonhomogeneous continuum, Bull. Acad. Polon. Sci. S´ er. Sci. Math. 28 (1980), no. 11–12, 627–630 (1981). [8] K. Kuperberg, W. Kuperberg, and W. R. R. Transue, On the 2-homogeneity of Cartesian products, Fund. Math. 110 (1980), no. 2, 131–134. [9] W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), no. 2, 361–394. [10] J. C. Mayer, L. G. Oversteegen, and E. D. Tymchatyn, The Menger curve. Characterization and extension of homeomorphisms of non-locally-separating closed subsets, Dissertationes Math. 252 (1986), 45 pp. [11] S. Mazurkiewicz, Sur les continus homog` enes, Fund. Math. 5 (1924), 137–146. [12] J. van Mill and G. M. Reed (eds.), Open problems in topology, North-Holland, Amsterdam, 1990. [13] M. W. Mislove and J. T. Rogers, Jr., Local product structures on homogeneous continua, Topology Appl. 31 (1989), no. 3, 259–267. [14] M. W. Mislove and J. T. Rogers, Jr., Addendum: “Local product structures on homogeneous continua”, Topology Appl. 34 (1990), no. 2, 209. ˇcepin, On 1-cycles and the finite dimensionality [15] W. J. R. Mitchell, D. Repovˇs, and E. V. Sˇ of homology 4-manifolds, Topology 31 (1992), no. 3, 605–623. [16] L. G. Oversteegen and J. R. Prajs, On confluently graph-like compacta, Fund. Math. 178 (2003), no. 2, 109–127. [17] J. R. Prajs, A homogeneous arcwise connected non-locally-connected curve, Amer. J. Math. 124 (2002), no. 4, 649–675. [18] J. R. Prajs, Filament local product structures in homogeneous continua, 2005, Preprint. [19] J. R. Prajs, Co-filament subcontinua in homogeneous spaces, 2006, Preprint. [20] J. R. Prajs and K. Whittington, Filament sets and homogeneous continua, 2005, Preprint. [21] J. R. Prajs and K. Whittington, Filament additive homogeneous continua, 2006, Preprint. To appear in Indiana Univ. Math. J. [22] J. R. Prajs and K. Whittington, Filament sets, aposyndesis, and the decomposition theorem of Jones, Trans. Amer. Math. Soc. (2006), Article in press. [23] J. T. Rogers, Jr., Classifying homogeneous continua, Topology Appl. 44 (1992), 341–352. [24] J. T. Rogers, Jr., Higher dimensional aposyndetic decompositions, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3285–3288. [25] W. Sierpi´ nski, Sur une propri´ et´ e topologique des ensembles d´ enombrables denses en soi, Fund. Math. 1 (1920), 11–28. [26] G. S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 393–400. [27] K. Whittington, Open connected sets in homogeneous spaces, Houston J. Math. 27 (2001), no. 3, 523–531.


Problems about the uniform structures of topological groups Ahmed Bouziad and Jean-Pierre Troallic

1. Introduction In the introduction of their fundamental paper entitled “Pseudocompactness and uniform continuity in topological groups” published in 1966, W.W. Comfort and K.A. Ross asserted, without proving it, that if every real-valued left uniformly continuous function on a topological group G is right uniformly continuous, then the left and right uniform structures on G coincide. Let us recall that the family of all {(x, y) ∈ G × G : x−1 y ∈ V }, with V a neighborhood of the identity element e in G, is a basis of the left uniform structure LG on G; a basis of the right uniform structure RG on G is obtained by replacing “x−1 y” by “xy −1 ”. Actually, forty years later, and despite many mathematicians’ efforts, it still isn’t known whether this property is true or false. The aim of this paper is to take stock of this problem, to present a few new ideas in order to study it, and to raise certain questions connected with the subject. The well-known class of all (Hausdorff) balanced topological groups is denoted by [SIN]. A topological group G is said to be balanced (or a [SIN]-group) if LG = RG , or, equivalently, if every neighborhood of the identity element contains a neighborhood which is invariant under all inner automorphisms of G. (Cf. for instance [37].) Following Protasov [35], we will say that G is functionally balanced (or an [FSIN]-group) if every bounded real-valued left uniformly continuous function on G is right uniformly continuous, and we will say that G is strongly functionally balanced if every real-valued left uniformly continuous function on G is right uniformly continuous (Itzkowitz [23]). The symbol [FSIN] (respectively [SFSIN]) stands for the class of all functionally (respectively strongly functionally) balanced topological groups. It is plain that [SIN] is a subclass of [SFSIN] and that [SFSIN] is a subclass of [FSIN]. All the questions below are motivated by the following “Itzkowitz’s Problem”, which was first raised by Itzkowitz in [21]: Question 1. Is [SFSIN] = [SIN]?

821?

In fact, we will especially consider the following bounded version of the problem (and from now on, the phrase “the main problem” will denote this case): Question 2. Is [FSIN] = [SIN]?

822?

Let us recall that between 1988 and 1992, Itzkowitz [21], Milnes [33] and Protasov [35] provided independent proofs that every locally compact functionally balanced topological group is balanced; in 1997, and in another direction, Megrelishvili, Nickolas and Pestov [32] proved that every locally connected functionally 359

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§38. Bouziad and Troallic, Uniform structures of topological groups

balanced topological group is balanced. Improvements of these results were presented by Itzkowitz [22] in his survey on the subject published in 1998. We will, of course, highlight the progress made after this date. 2. The two versions of Itzkowitz’s problem Considering a bounded version and an unbounded version of the problem was not immediately deemed useful; for example, in the locally compact case in [21], or in the locally connected case in [32], it was stated that any G in [SFSIN] is balanced, although the proof works with [FSIN] instead of [SFSIN]. Nevertheless, it remains possible that the two problems are only one. 823?

Question 3. Is [SFSIN] = [FSIN]? Clearly, a negative answer to Question 3 would imply a very strong negative answer to Question 2. In 1991, Protasov [35] gave a positive answer to Question 2 for the class of almost metrizable groups (a class which contains both that of locally compact groups and of metrizable groups) by using the following very interesting characterization of [FSIN]: let VG (e) denote the neighborhood system of the identity element e of a given topological group G; then G is a member of [FSIN] if and only if G satisfies the Protasov and Saryev criterion [36], that is to say, if and only if for all A ⊂ G and V ∈ VG (e), there is U ∈ VG (e) such that U A ⊂ AV . The part played by this criterion in the problem was made completely clear in [4, 6] when observing, after an immediate re-writing, that it means the equality of the proximities on G induced by the left and right uniformities. Another way to formulate Protasov and Saryev’s criterion consists in saying that for each A, B ⊂ G and V ∈ VG (e) such that AV ⊂ B, there is U ∈ VG (e) such that U A ⊂ B. This formulation leads us to propose the following criterion for G to be strongly functionally balanced. It is easily derivable from the work of Leader [31]. To state it, the following terminology is needed. A sequence (An )n∈N of subsets of G is said to be strongly left increasing (respectively strongly right increasing) if there is V ∈ VG (e) such that An V ⊂ An+1 (respectively V An ⊂ An+1 ) for each n ∈ N. Proposition. The following conditions are equivalent for any topological group G. (1) G is strongly functionally balanced. (2) Every sequence of subsets of G which is strongly left increasing is strongly right increasing. A topological group is said to be non-Archimedean if there exists a base for the neighborhood system of the identity element which consists of open subgroups. It was discovered by Hernández [13] that every G in [SFSIN] which is non-Archimedean and ℵ0 -bounded is balanced. This result was recently extended in [40] to every non-Archimedean group which is strongly functionally generated by the set of all its subspaces of countable o-tightness. For all we know, these

Some remarks about [FSIN] and [SIN]

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are the only instances where unbounded uniformly continuous functions were really involved. Let us take the opportunity to state the criterion for balancedness that ndez established beforehand: A topological group G is balanced (i.e., T Herná−1 T ∈ VG (e) for all V ∈ VG (e)) if and only if a∈A aV a−1 ∈ VG (e) for x∈G xV x every left uniformly discrete subset A of G and every V ∈ VG (e). This very important criterion was already implicitly used in [32], and explicitly formulated in [22]. Let us recall that a subset A of G is said to be left uniformly discrete if there is V ∈ VG (e) such that aV and bV are disjoint whenever a, b ∈ A and a 6= b. From now on, we will only consider the bounded version of Itzkowitz’s Problem, it being understood that most of the problems raised below obviously admit an unbounded version. 3. Some remarks about [FSIN] and [SIN] Let G be a topological group. If (and only if) for any precompact uniform space Y , every left uniformly continuous mapping of G into Y is right uniformly continuous, then G ∈ [FSIN]. If Y runs through the larger class of all bounded uniform spaces, a characterization of [SIN] is obtained. Before specifying this point, let us recall some definitions and properties. A uniform space Y is said to be bounded if all real-valued uniformly continuous functions on Y are bounded. Another concept in its right place here is that of injective uniform space: Y is said to be injective if whenever A is a subspace of a uniform space X, any uniformly continuous mapping of A into Y has a uniformly continuous extension to X [18]. It can be shown that if Y is injective, then Y is bounded. The most familiar example of an injective uniform space is that of the unit interval [0, 1]; this fact, proved by Katˇetov [27, 28], is here of great significance since the belonging of the topological group G to [FSIN] means that the left and right uniformities on G induce the same proximity on G. Another standard injective uniform space is the metric Hedgehog H(A) over a set A, that is the set of all (a, x) (a ∈ A, 0 ≤ x ≤ 1), A × {0} being reduced to a point, with the metric d((a, x), (a, y)) = |x − y| and d((a, x), (b, y)) = x + y if a 6= b. A family (Ai )i∈I of subsets of G is said to be left uniformly discrete if there is V ∈ VG (e) such that Ai V and Aj V are disjoint whenever i, j ∈ I and i 6= j. As already said in Section 2, a subset A of G is left uniformly discrete if the family ({a}) T a∈A is left uniformly discrete. The subset A of G is said to be right thin (in G) if a∈A aV a−1 is a neighborhood of the identity element e for every V ∈ VG (e). Right uniform discreteness and left thinness are defined similarly. Finally, the subset A of G is said to be lower uniformly discrete if there is V ∈ VG (e) such that V aV and V bV are disjoint whenever a, b ∈ A and a 6= b. The following is a key lemma; knowing whether it can be extended to any left uniformly discrete subset A of G is equivalent to Question 2 since, as said in Section 2, if every left uniformly discrete subset of G is right thin, then G is balanced [22]. Lemma ([5]). Every lower uniformly discrete subset of a functionally balanced group G is right thin in G.

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Proposition. Let G be a topological group. Then the following are equivalent: (1) G is balanced. (2) For any bounded uniform space Y , every left uniformly continuous mapping f : G → Y is right uniformly continuous. (3) For any injective uniform space Y , every left uniformly continuous mapping f : G → Y is right uniformly continuous. (4) For any set A, every left uniformly continuous mapping f : G → H(A) is right uniformly continuous. (5) Any left uniformly discrete family of subsets of G is right uniformly discrete. Proof. Obviously, (1) implies (2). Any injective uniform space being bounded, (2) implies (3). Since H(A) is injective, (3) implies (4). The implication (4) ⇒ (5) holds for any two uniformities on a given set X (in place of the left and right uniformities on G); see [8] or [30]. Finally, let us suppose that (5) holds; then any two subsets of G which are left proximal are right proximal (so that G ∈ [FSIN]), and every left uniformly discrete subset of G is lower uniformly discrete; therefore, by the above key lemma, (1) is satisfied. (Note that the equivalence between (1), (2) and (3) is also a consequence of the well-known fact that every uniform space can be embedded in an injective uniform space [18].) In view of the previous proposition, Question 2 could be stated as follows: Question 4. Let G be a functionally balanced group. Is any left uniformly discrete family of subsets of G right uniformly discrete? Let us say that a topological group G is injective (respectively bounded) if the uniform space (G, LG ) (or, equivalently, (G, RG )) is injective (respectively bounded). Since any injective uniform space is proximally fine [17], the answer to Question 2 is positive for all injective topological groups. Proposition. Every injective topological group which is functionally balanced is balanced. 824?

Question 5. Is it true, more generally, that every bounded member of [FSIN] belongs to [SIN]? It should be pointed out that the answer to the following question is not clear.

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Question 6. Is every injective (respectively bounded) topological group functionally balanced? 4. The class [ASIN] A natural approach to the main problem is to consider any class [C] of topological groups which contains the class [SIN] as closely as possible, and try to prove that [C] also includes the class [FSIN]. The dual problem consists in examining whether the inclusion [C] ∩ [FSIN] ⊂ [SIN] holds, the class [C] now being as wide as possible. To illustrate that idea, let us consider the class [ASIN] of all almost

A few other questions

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balanced topological groups, that is the class of all topological groups G for which the identity element e has at least a right thin neighborhood in G. Clearly, the class [ASIN] is much larger than [SIN]. The following is established in [5]. Theorem. [ASIN] ∩ [FSIN] = [SIN]. In view of that property, Question 2 becomes: Is [FSIN] ⊂ [ASIN]? Moreover, for any class of topological groups contained in [ASIN], a positive answer to the main problem holds; it applies, for instance, to every topological group which is locally precompact or which contains an open subgroup belonging to [SIN]. Here is another interesting class contained in [ASIN]: Proposition. Let G be a topological group. Suppose that the identity element of G has a neighborhood V such that every bounded real-valued continuous function on G is left uniformly continuous, when restricted to V . Then G is a member of [ASIN]. It is proved by Itzkowitz and Tkachuk in [26] that every uniformly functionally complete topological group G is balanced; of course, this follows from the previous proposition. Recall that G is said to be uniformly functionally complete [26], or with property U [29], if every real-valued (or, equivalently [7], every bounded realvalued) continuous function on G is left uniformly continuous. 5. A few other questions In this section we collect some concrete open questions related to the main problem. Question 7. Let G be a functionally balanced group. Is every left uniformly discrete subset of G right uniformly discrete?

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Question 8. Let G be a functionally balanced group. Is every left precompact subset of G right precompact?

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Question 9. Let G be a functionally balanced group. Let us suppose that every left precompact subset of G is right precompact. Is G balanced then?

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The main statement of a recent paper by Itzkowitz [24] consists in saying that if G ∈ [FSIN] and if every left uniformly discrete subset of G is right uniformly discrete, then G ∈ [SIN]. This is to be compared with the implication (5) ⇒ (1) in the first proposition of Section 3 above. We must admit that one point in Itzkowitz’s argumentation eluded us. It is well known that any balanced topological group is isomorphic (topologically and algebraically) to a subgroup of a product of balanced metrizable topological groups [9]. (See [37] for details.) This suggests the two following questions: Question 10. Is every member of [FSIN] isomorphic to a subgroup of a product of metrizable topological groups? The dual question is:

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Question 11. Let G be a member of [FSIN] which is isomorphic to a subgroup of a product of metrizable topological groups. Is G balanced? A topological group G is said to be ℵ0 -bounded if for every neighborhood V of the identity element, there is a countable subset A of G such that G = AV . It is well known that G is ℵ0 -bounded if and only if it is isomorphic to a subgroup of a product of metrizable separable topological groups [10].

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Question 12. Let G be a member of [FSIN] which is ℵ0 -bounded. Is G balanced? In fact, the question arises even in the following simple case.

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Question 13. Is every countable member of [FSIN] balanced? Let us remark that a positive answer to Question 13 will imply a positive answer to Question 4 for ℵ0 -bounded groups. 6. A representative case The fact that a metrizable topological group which is functionally balanced is balanced belongs to the theory of uniform spaces. The same remark holds, more generally, for every topological group such that the neighborhood system of the identity element has a linearly ordered base, and this follows from the proximally fineness (proved in [1]) of every uniform space (X, U) such that U has a linearly ordered base. The above positive answer to Question 2 for the class of all injective topological groups rested on the same sort of argument. The following combinatorial lemma, established in [1], is essential for the approach of these results. Lemma. Let (X, U) be a uniform space and W ∈ U a symmetric entourage. Let us suppose that (xα , yα ) 6∈ W 3 for all α ∈ Γ, where Γ is a set of regular cardinal, and xα , yα ∈ X. Then, there is a set Γ0 ⊂ Γ with the same cardinal as Γ such that (xα , yβ ) 6∈ W for all α, β ∈ Γ0 . Surprisingly enough, in the context of topological groups, and in order to tackle our main problem, that lemma may be used under very general topological conditions. Let us illustrate this by the proposition below. Let us say that a subset Y of a topological group G is relatively o-radial in G ifSfor every y ∈ YSand every family (Oi )i∈I of open subsets of G such that y ∈ cl( i∈I Oi ∩ Y ) \ i∈I cl(Oi ), there is a set J ⊂ I of regular cardinality such that for every neighborhood V of y in G, we have |{j ∈ J : Oj ∩ V = ∅}| < |J|. Obviously, we will say that G is o-radial if it is relatively o-radial in itself. Radial spaces are defined in [15] (cf. also [2]); every radial topological group is o-radial. If Y is relatively o-Malykhin in G (as defined in [6]), that is for instance the case if Y is left (or right) precompact [6], then Y is relatively o-radial in G. All locally precompact topological groups, all q-groups (as defined in [12]), are o-Malykhin (i.e., relatively o-Malykhin in themselves), and therefore o-radial. Proposition. Every functionally balanced o-radial topological group is balanced.

References

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Proof. Suppose that G is not balanced. Then there is a symmetric open −1 W ∈ VG (e) such that for every U ∈ VG (e) one can find gU , hU ∈ G with gU hU ∈ U S −1 −1 6 and gU hU 6∈ W . Clearly, e ∈ cl( U ∈VG (e) gU W hU ). Since G is o-radial, there S −1 is Γ ⊂ VG (e) such that the cardinal of Γ is regular and e ∈ cl( U ∈Γ0 gU W hU ) for 0 every Γ ⊂ Γ with the same cardinal as Γ. By the previous lemma, there is a set 2 00 Γ00 ⊂ Γ with the same cardinal as Γ such that gU h−1 V 6∈ W for all U, V ∈ Γ . Let 00 00 us put A = {gU : U ∈ Γ } and B = W {hV : V ∈ Γ }; then A and B are left, but not right, proximal which contradicts the functional balance of G. In fact, if G is an o-radial topological group, then the uniform space (G, LG ) is proximally fine, and that remains true if G is more generally strongly functionally generated (in the sense of [3]) by its relatively o-radial subspaces. References [1] E. M. Alfsen and O. Njastad, ˙ Totality of uniform structures with linearly ordered base, Fund. Math. 52 (1963), 253–256. [2] A. V. Arhangel0 ski˘ı, Some properties of radial spaces, Mat. Zametki 27 (1980), no. 1, 95–104, Translation: Math. Notes 27 (1980) no. 1, 50–54. [3] A. V. Arhangel0 ski˘ı, Topological function spaces, Kluwer Academic Publishers, Dordrecht, 1992. [4] A. Bouziad and J. P. Troallic, Functional equicontinuity and uniformities in topological groups, Topology Appl. 144 (2004), 95–107. [5] A. Bouziad and J.-P. Troallic, Nonseparability and uniformities in topological groups, Topology Proc. 28 (2004), no. 2, 343–359. [6] A. Bouziad and J. P. Troallic, Left and right uniform structures on topological groups, Topology Appl. 153 (2006), no. 13, 2351–2361. [7] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483–496. [8] Z. Frol´ık, Basic refinements of the category of uniform spaces, TOPO 72—general topology and its applications, Lecture Notes in Math., vol. 378, Springer, Berlin, 1974, pp. 140–158. [9] M. I. Graev, Theory of topological groups. I. Norms and metrics on groups. Complete groups. Free topological groups, Uspekhi Mat. Nauk 5 (1950), no. 2(36), 3–56, Translation: Free topological groups, Amer. Math. Soc. Transl. 35 (1951). [10] I. I. Guran, Topological groups similar to Lindel¨ of groups, 1981, Translation: Soviet Math. Dokl. 23 (1981) 173–175, pp. 1305–1307. [11] G. Hansel and J.-P. Troallic, Equicontinuity, uniform structures and countability in locally compact groups, Semigroup Forum 42 (1991), no. 2, 167–173. [12] G. Hansel and J.-P. Troallic, Sequential criteria for the equality of uniform structures in q-groups, Topology Appl. 57 (1994), no. 1, 47–52. [13] S. Hern´ andez, Topological characterization of equivalent uniformities in topological groups, Topology Proc. 25 (2000), Spring, 181–188. [14] S. Hern´ andez, Uniformly continuous mappings defined by isometries of spaces of bounded uniformly continuous functions, Houston J. Math. 29 (2003), no. 1, 149–155. [15] H. Herrlich, Quotienten geordneter R¨ aume und Folgenkonvergenz, Fund. Math. 61 (1967), 79–81. [16] S. Hingano, On uniformities and uniformly continuous functions on factor-spaces of topological groups, Topology Proc. 29 (2005), no. 2, 521–526. [17] M. Huˇsek, Factorization of mappings (products of proximally fine spaces), Seminar Uniform ´ ˇ Spaces (Prague, 1973–1974), Mat. Ustav Ceskoslovenskˇ e Akad. V´ ed, Prague, 1975, pp. 173– 190.

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[18] J. R. Isbell, Uniform spaces, American Mathematical Society, Providence, RI, 1964. [19] G. Itzkowitz, Continuous measures, Baire category, and uniform continuity in topological groups, Pacific J. Math. 54 (1974), no. 2, 115–125. [20] G. Itzkowitz, Uniform structure in topological groups, Proc. Amer. Math. Soc. 57 (1976), no. 2, 363–366. [21] G. Itzkowitz, Uniformities and uniform continuity on topological groups, General topology and applications (Staten Island, NY, 1989), Lecture Notes in Pure and Appl. Math., vol. 134, Marcel Dekker Inc., New York, 1991, pp. 155–178. [22] G. Itzkowitz, On balanced topological groups, Topology Proc. 23 (1998), Spring, 219–233. [23] G. Itzkowitz, Projective limits and balanced topological groups, Topology Appl. 110 (2001), no. 2, 163–183. [24] G. Itzkowitz, Functional balance, discrete balance, and balance in topological groups, Topology Proc. 28 (2004), no. 2, 569–577. [25] G. Itzkowitz, S. Rothman, H. Strassberg, and T. S. Wu, Characterization of equivalent uniformities in topological groups, Topology Appl. 47 (1992), no. 1, 9–34. [26] G. Itzkowitz and V. V Tkachuk, Uniformly functionally complete and uniformly R-factorizable groups, Preprint. [27] M. Katˇ etov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85–91. [28] M. Katˇ etov, Correction to “On real-valued functions in topological spaces” (Fund. Math. 38 (1951) 85–91), Fund. Math. 40 (1953), 203–205. [29] J. M. Kister, Uniform continuity and compactness in topological groups, Proc. Amer. Math. Soc. 13 (1962), 37–40. [30] M. Kosina and P. Pt´ ak, Intrinsic characterization of distal spaces, Seminar Uniform Spaces ´ ˇ (Prague, 1973–1974), Mat. Ustav Ceskoslovenskˇ e Akad. V´ ed, Prague, 1975, pp. 217–231. [31] S. Leader, Spectral structures and uniform continuity, Fund. Math. 60 (1967), 105–115. [32] M. Megrelishvili, P. Nickolas, and V. Pestov, Uniformities and uniformly continuous functions on locally connected groups, Bull. Austral. Math. Soc. 56 (1997), no. 2, 279–283. [33] P. Milnes, Uniformity and uniformly continuous functions for locally compact groups, Proc. Amer. Math. Soc. 109 (1990), no. 2, 567–570. [34] W. H. Previts and T. S. Wu, Notes on balanced groups, Bull. Inst. Math. Acad. Sinica 29 (2001), no. 2, 105–123. [35] I. V. Protasov, Functionally balanced groups, Mat. Zametki 49 (1991), no. 6, 87–91, Translation: Math. Notes 49 (1991) no. 6, 614–616. [36] I. V. Protasov and A. Saryev, The semigroup of closed subsets of a topological group, Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.-Tekhn. Khim. Geol. Nauk (1988), no. 3, 21–25. [37] W. Roelcke and S. Dierolf, Uniform structures in topological groups and their quotients, McGraw-Hill, New York, 1981. [38] J. P. Troallic, Sequential criteria for equicontinuity and uniformities on topological groups, Topology Appl. 68 (1996), no. 1, 83–95. [39] J. P. Troallic, Equicontinuity, uniform continuity and sequences in topological groups, Topology Appl. 93 (1999), no. 3, 179–190. [40] J. P. Troallic, Equicontinuity and balanced topological groups, Topology Appl. 135 (2004), 63–71.


On some special classes of continuous maps Maria Manuel Clementino and Dirk Hofmann 1. Special morphisms of Top Triquotient maps were introduced by E. Michael in [24] as those continuous maps f : X → Y for which there exists a map ( )] : OX → OY such that, for every U, V in the lattice OX of open subsets of X: (T1) U ] ⊆ f (U ), (T2) X ] = Y , (T3) U ⊆ V ⇒ U ] ⊆ V ] , S (T4) (∀y ∈ U ] ) (∀Σ ⊆ OX directed) f −1 (y) ∩ U ⊆ Σ ⇒ (∃S ∈ Σ) y ∈ S ] . It is easy to check that, if f : X → Y is an open surjection, then the direct image f ( ) : OX → OY satisfies (T1)–(T4). If f : X → Y is a retraction, so that there exists a continuous map s : Y → X with f ◦ s = 1Y , then ( )] := s−1 ( ) satisfies (T1)–(T4). Moreover, if f : X → Y is a proper surjection (by proper map we mean a closed map with compact fibres: see [2]), then U ] := Y \ f (X \ U ) fulfills (T1)– (T4). That is, open surjections, retractions and proper surjections are triquotient maps. But there are triquotient maps which are neither of these maps (cf. [3, 15] for examples). However, we do not know whether these three classes of maps describe completely triquotient maps, in the sense we describe now: Question 1. Is it true that any triquotient map can be factored through (possibly infinitely many) open surjections, proper surjections and retractions?

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T. Plewe in [26] related triquotient maps to Topological Grothendieck Descent Theory (see [17]). We recall that a continuous map f : X → Y is an effective descent map if its pullback functor f ∗ : Top/Y → Top/X, that assigns to each g : W → Y its pullback along f , is monadic. If f ∗ is premonadic, then f is a descent map (see [18, 19]). Descent maps are exactly universal quotient maps [10], or pullback-stable quotient maps, that is quotient maps whose pullback along any map is still a quotient. We point out here that this class of maps was introduced independently by B. Day and M. Kelly [10], by E. Michael [23], under the name biquotient maps, and by O. H´ ajek [11], as limit lifting maps. Effective descent maps turned out to be very difficult to describe topologically. The only characterisation that is known is due to J. Reiterman and W. Tholen [27] and uses heavily ultrafilter convergence. (We will concentrate on ultrafilter convergence later in this work.) Problem 2. Find a characterisation of topological effective descent maps in terms of the topologies or the Kuratowski closures. The authors acknowledge partial financial assistance by Centro de Matem´ atica da Universidade de Coimbra/FCT and Unidade de Investiga¸c˜ ao e Desenvolvimento Matem´ atica e Aplica¸c˜ oes da Universidade de Aveiro/FCT. 367

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§39. Clementino and Hofmann, On some special classes of continuous maps

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One possible approach to this problem could be via the existence of a map between the topologies that resembles the map ( )] introduced by E. Michael to define triquotient maps. Indeed, it is interesting to notice that one can characterise several of these classes of morphisms using a map OX → OY as follows: a continuous map f : X → Y is: (1) a universal quotient map (= biquotient = limit lifting = descent) if and (T1)–(T3) and only if there exists a map ( )] : OX → OY satisfying S (U4) (∀y ∈ Y ) (∀Σ ⊆ OX directed) f −1 (y) ⊆ Σ ⇒ (∃S ∈ Σ) y ∈ S ] . (2) a proper surjection if and only if there exists a map ( )] : OX → OY satisfying (T1)–(T3) and, for every U ∈ OX S (P4) (∀y ∈ U ] ) (∀Σ ⊆ OX directed) f −1 (y) ∩ U ⊆ Σ ⇒ (∃S ∈ Σ) y ∈ S ] and f −1 (y) ⊆ S. (3) an open surjection if and only if there exists a map ( )] : OX → OY satisfying (T1)–(T3) and, for every U ∈ OX S (O4) (∀y ∈ f (U )) (∀Σ ⊆ OX directed) f −1 (y) ∩ U ⊆ Σ ⇒ (∃S ∈ ] Σ) y ∈ S . 836?

Problem 3. Describe effective descent maps via the existence of a map ( )] similar to those described above. A possible approach to Problem 3 might be making use of the following result, that we could prove only for maps between finite topological spaces. It is based on the existence of a map between the lattices of locally closed subsets (i.e., the subsets which are an intersection of an open and a closed subset), which we will denote by LC( ). Theorem 1. If X and Y are finite spaces, a continuous map f : X → Y is an effective descent map if and only if, for every pullback g : W → Z of f , there exists a map ( )] : LC(W ) → LC(Z) such that, for every A, B ∈ LC(W ), (1) (2) (3) (4)

A] ⊆ g(A); W ] = Z; A ⊆ B ⇒ A] ⊆ B ] ; (∀z ∈ Z) g −1 (z) ⊆ A ⇒ z ∈ A] .

We believe that the work of G. Richter [28] may be inspiring to attack Problem 3. In the pioneer work [15], G. Janelidze and M. Sobral describe several classes of maps using convergence, whenever X and Y are finite spaces. Theorem 2 ([15]). If X and Y are finite spaces, a continuous map f : X → Y is: (a) a universal quotient if and only if, for every y0 , y1 ∈ Y with y1 → y0 , there exist x0 , x1 ∈ X such that x1 → x0 , f (x0 ) = y0 and f (x1 ) = y1 ; (b) an effective descent map if and only if, for every y0 , y1 , y2 ∈ Y with y2 → y1 → y0 , there exist x0 , x1 , x2 ∈ X such that x2 → x1 → x0 and f (xi ) = yi , for i = 0, 1, 2;

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(c) a triquotient map if and only if, for every y0 , . . . , yn ∈ Y with yn → · · · → y0 , there exist x0 , . . . , xn ∈ X such that xn → · · · → x0 and f (xi ) = yi , for i = 0, . . . , n. (d) a proper map if and only if, for every x1 ∈ X and y0 ∈ Y with f (x1 ) → y0 , there exists x0 ∈ X such that x1 → x0 and f (x0 ) = y0 ; (e) an open map if and only if, for every x0 ∈ X and y1 ∈ Y with y1 → f (x0 ), there exists x1 ∈ X such that x1 → x0 and f (x1 ) = y1 ; (f) a perfect map if and only if, for every x1 ∈ X and y0 ∈ Y with f (x1 ) → y0 , there exists a unique x0 ∈ X such that x1 → x0 and f (x0 ) = y0 ; (g) a local homeomorphism (or étale map) if and only if, for every x0 ∈ X and y1 ∈ Y with y1 → f (x0 ), there exists a unique x1 ∈ X such that x1 → x0 and f (x1 ) = y1 . We recall that a continuous map f : X → Y is perfect if it is proper and Hausdorff (i.e., if f (x) = f (x0 ) and x 6= x0 , there exists U, V ∈ OX with x ∈ U , x0 ∈ V and U ∩ V = ∅), and that it is a local homeomorphism, or an étale map, if it is open and, for each x ∈ X, there exists U ∈ OX such that x ∈ U and fU : U → f (U ) is a homeomorphism. This work led us to investigate the extension of these characterisations to maps between (infinite) topological spaces. The right setting to use convergence turned out to be the ultrafilter convergence. As a side result we also obtained, together with W. Tholen, a useful characterisation of exponentiable maps via convergence (see [9]) we will mention in Section 2. The results corresponding to (a), (d), (e), (f) were either known or easy to obtain; in fact, the characterisation of universal quotient maps using convergence is the basis for the definition of limit lifting maps by Hájek, and the descriptions of open, proper and perfect maps are straightforward (see [4]). Statement (b) corresponds, in the infinite case, to the Reiterman–Tholen characterisation of effective descent maps. Indeed, although this is not completely evident in the original formulation [27], the notions and techniques introduced in [4] clarify completely the analogy between these characterisations. In the latter paper, we also generalized the Janelidze–Sobral–Clementino characterisation of triquotient maps (c) to the infinite case, as we will explain later. After that, only a characterisation of local homeomorphism, using ultrafilter convergence, remained unknown to us. To explain the problem we first state the characterisation of proper, perfect and open maps. Proposition 3. (1) A continuous map f : X → Y is proper (perfect) if and only if, for each ultrafilter x in X with f [x] → y in Y , there exists a (unique) x in X such that x → x and f (x) = y: _ _ _/ x X _ _x f / y f [x] Y

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(2) A continuous map f : X → Y is open if and only if, for each x ∈ X and each ultrafilter y in Y with y → f (x) in Y , there exists an ultrafilter x in X such that x → x and f (x) = y: X f

Y

_ _ _/ x _ _x / f (x) y

Analysing the characterisations of proper and perfect map, we conjectured that, from the characterisation of open map, one could obtain a characterisation of local homeomorphism imposing the unicity of the lifting of the convergence y → f (x). Indeed, the parallelism between the two situations becomes evident once we observe that, if we denote by δf : X → X ×Y X, x 7→ (x, x), the continuous map induced by the pullback property of the (pullback) diagram below 1

X X KK KKδf KK % X ×Y X

1X

π2

π1

( X

f

! /X /Y

f

then: a continuous map f : X → Y is perfect if and only if f and δf are proper maps; a continuous map f : X → Y is a local homeomorphism if and only if f and δf are open maps. Eventually we have shown, together with G. Janelidze [8], that our conjecture was wrong. Calling a continuous map f : X → Y having the unicity of the lifting of y → f (x) described above a discrete fibration (using the parallelism with categorical discrete fibrations), one has: Proposition 4 ([8]). Every local homeomorphism is a discrete fibration (and the converse is false). We could prove that the two notions coincide under some conditions on the domain of the map. For that, given a cardinal number λ, we call a topological space X a λ-space if the character of X is at most λ and each subset of X with cardinality less than λ is closed. Theorem 5 ([8]). If X is a λ-space, for some cardinal λ, then, for continuous maps with domain X, local homeomorphisms and discrete fibrations coincide. Among λ-spaces one has the indiscrete spaces (= 0-spaces), the Alexandrov spaces (= 1-spaces) and the first countable T1 -spaces (= ℵ0 -spaces). 837?

Problem 4. Characterise those topological spaces X such that, for a continuous map f : X → Y , f is a local homeomorphism if and only if it is a discrete fibration.

Special morphisms of Top

371

In order to formulate more results and problems in this context we need to consider iterations of the ultrafilter convergence. The right way of doing this is making it a functorial process. There are two natural choices in this direction. We may use the ultrafilter functor U : Rel → Rel, which assigns to each set X its set of ultrafilters U X, and to each relation r : X 9 Y the corresponding relation U r : U X 9 U Y (see for instance [1]). For maps f : X → Y , U f : U X → U Y is the usual map; for simplicity we write U f (x) = f [x]. The most valuable functor in this study is the functor Conv : Top → URS, where URS is the category of ultrarelational spaces and convergence preserving maps (see [4]), which assigns to each topological space X the space Conv(X); here Conv(X) is the set consisting of pairs (x, x), where x is a point and x is an ultrafilter converging to x in X, equipped with a convergence structure as follows: first we consider the map p : Conv(X) → X with p(x, x) = x; an ultrafilter X converges to (x, x) in Conv(X) if p[X] = x. Each continuous map f : X → Y induces a map Conv(f ) : Conv(X) → Conv(Y ) with (x, x) 7→ (f [x], f (x)), which preserves the convergence structure (see [4] for details). It is clear that we can consider instead Conv : URS → URS. Furthermore, the map p : Conv(X) → X preserves the structure, so that it defines a natural transformation p : Conv → 1URS . This functor Conv is an excellent tool to describe our classes of maps via their lifting of convergence. For that we need to consider (possibly transfinite) iterations of the functor Conv : URS → URS, as described in [4]. For an ordinal number α we call a continuous map f : X → Y between ultrarelational (or topological) spaces α-surjective if, for every β < α, Convβ (f ) : Convβ (X) → Convβ (Y ) is surjective; f : X → Y is Ω-surjective if Convα (f ) is surjective for every ordinal α. Theorem 6 ([4]). For a continuous map f : X → Y between topological spaces, (1) f is 1-surjective if and only if it is surjective; (2) f is 2-surjective if and only if it is a universal quotient map (if and only if it is a descent map); (3) f is 3-surjective if and only if it is an effective descent map; (Ω) f is Ω-surjective if and only if it is a triquotient map. Similarly to Problem 2, we may formulate the following Problem 5. Study the properties of the classes of 4-surjective, . . . , n-surjective, ωsurjective maps, and possible characterisations of these classes using the topologies (or even the sequential closures). Concerning assertion (Ω) above, it is shown in [4] that, for a continuous map f : X → Y , f is a Ω-surjection if and only if f is a λY -surjection, where λY is the successor of the cardinal of Y . This covers the result already known for a continuous map between finite spaces: f is a triquotient (hence Ω-surjective) if and only if it is ω-surjective (see [3, 15]).

838?

372 839?


Problem 6. Characterise those topological spaces Y such that, for a continuous map f : X → Y , f is Ω-surjective if and only if it is ω-surjective. Using the functor U : Rel → Rel instead of Conv : URS → URS, one can also iterate U and formulate the notions of U -α-surjective map, for any continuous map f : X → Y between topological spaces. (We will keep the name α-surjective for Conv-α-surjective maps.) It is easy to check that every 3-surjective map is U -3-surjective, i.e., every effective descent map is U -3-surjective.

840?

Question 7. Is every U -3-surjective map an effective descent map?

841?

Question 8. If the answer to the previous question is negative, is the class of effective descent maps the least pullback-stable class containing the U -3-surjective maps? Furthermore, the functor Conv may be also useful to characterise local homeomorphisms as special discrete fibrations. For instance, one may ask the following:

842?

Question 9. Is every continuous map f such that both f and Conv(f ) are discrete fibrations a local homeomorphism? There is another problem in Topological Descent Theory, described in the sequel, that justifies the study of local homeomorphisms, or étale maps, using convergence. The notion of an effective (global-)descent map can be generalised by considering instead of all morphisms with codomain Y a well-behaved subclass E(Y ) of morphisms. One important example is obtained by taking E the class of all étale maps, so that E(Y ) is the category of étale bundles over the space Y . A continuous map f : X → Y is called an effective étale-descent map if the pullback functor f ∗ : E(Y ) → E(X) is monadic. For finite spaces X and Y , the problem of characterising effective étale-descent maps was solved by G. Janelidze and M. Sobral: Theorem 7 ([16]). The morphism f : X → Y in FinTop is an effective étaledescent map if and only if the functor ϕ : Z(Eq(f )) → Y is an equivalence of categories. Here Eq(f ) is the equivalence relation on X induced by f , and Z(Eq(f )) is the category having as objects the points of X; a morphism from x to x0 is an equivalence class of zigzags in X (see Figure 1). Now ϕ : Z(Eq(f )) → Y is an equivalence of categories if and only if (1) f : X → f (X) is a quotient map, (2) Z(Eq(f )) is a preorder and (3) f : X → Y is essentially surjective on objects (i.e., for every y ∈ Y there exists x ∈ X such that f (x) → y → f (x)). The obvious question is now how to transport this result into the context of all topological spaces.

Corresponding morphisms in related categories

373

x

xn−1 ∼f x0n−1 xn−2 ∼f x0n−2 x1 ∼f x01 x0 . Figure 1. An equivalence class of zigzags Question 10. Characterise effective étale-descent maps f : X → Y between arbitrary topological spaces. A possible solution to the problem above requires most likely translations of point-convergence notions and arguments to (ultra)filter-convergence ones. Both notions, of local homeomorphism and quotient map, should be considered in this problem via ultrafilter convergence. We have already mentioned the study of local homeomorphisms using convergence developed in [8]; possible descriptions of quotients, and their relations to zigzags, are studied in [13]. 2. Corresponding morphisms in related categories We study now the same problems in categories related to Top. Here same will be used with two meanings: either we consider characteristic categorical properties of the morphisms, or we deal with topological categories whose objects and morphisms have a description similar to the (ultrafilter) convergence description of Top. (We will not focus in this latter subject since it would take us too far from our purpose here. But we refer the reader to [5].) We start by considering some important supercategories (improvements) of Top. The interest in these categories has its roots in the fact that many categorically defined constructions either cannot be carried out in Top or destroy properties of spaces or maps. In order to perform these constructions, topologists move (temporarily) outside Top into larger but better behaved environments such as the category PsTop of pseudotopological spaces and continuous (i.e., convergence preserving) maps. Recall that a pseudotopology on a set X may be described as a convergence relation x → x between ultrafilters x on X and points x ∈ X such that the principal ultrafilter x˙ converges to x. The category PsTop contains Top as a full and reflective subcategory; in fact, it is the quasitopos hull of Top (see [12] for details). Being in particular a quasitopos, PsTop is locally cartesian closed and therefore the class of effective descent morphisms coincides with the class of quotient maps.

843?

374


A pseudotopology on aTset X is called a pretopology if it is closed under intersections in the sense that y→x y ⊆ x implies x → x, for each ultrafilter x ∈ U X and each x ∈ X. Hence Tconvergence to a point x is completely determined by the neighbourhood filter y→x y. Together with continuous maps pretopological spaces form the category PrTop. In [12] is is shown that PrTop is the extensional topological hull of Top, that is, the smallest extensional topological category containing Top nicely. However, in contrary to PsTop, the category PrTop is not cartesian closed. Exponentiable pretopological spaces are characterised in [22] as those spaces where each point has a smallest neighbourhood. The map version of this result is established in [29]: it states that a continuous map f : X → Y between pretopological spaces is exponentiable if and only if each x ∈ X has a neighbourhood V such that, for each ultrafilter x in X, if V ∈ x and f [x] → f (x) in Y , then x → x in X. Whereas exponentiable objects and morphisms are fully understood in PrTop, effective descent maps have not been described yet. 844?

Question 11. Characterise effective descent maps f : X → Y between pretopological spaces. The study of these classes of maps is also interesting in metric-like structures. Together with metric spaces we also consider premetric spaces. By a premetric space we mean a set X together with a map a : X ×X → [0, ∞] such that a(x, x) = 0 and a(x, z) ≤ a(x, y) + a(y, z), for any x, y, z ∈ X; that is, a premetric is a, possibly infinite, reflexive and not necessarily non-symmetric distance. We consider now the categories Met, of metric spaces and non-expansive maps, and PMet, of premetric spaces and non-expansive maps. Exponentiable and effective descent maps between metric, and more generally premetric, spaces are characterised in [7] and [6] respectively. We list here the results which might serve, together with the corresponding results for topological spaces, as a guideline for the study of these classes of maps in approach spaces as outlined below. Theorem 8. A non-expansive map f : (X, a) → (Y, b) between premetric spaces is exponentiable in PMet if and only if, for each x0 , x2 ∈ X, y1 ∈ Y and u0 , u1 ∈ R such that u0 ≥ b(f (x0 ), y1 ), u1 ≥ b(y1 , f (x2 )) and u0 + u1 = max{a(x0 , x2 ), b(f (x0 ), y1 ) + b(y1 , f (x2 ))} < ∞, (∀ε > 0) (∃x1 ∈ f −1 (y1 )) a(x0 , x1 ) < u0 + ε and a(x1 , x2 ) < u1 + ε. Theorem 9. A non-expansive map f : (X, a) → (Y, b) between metric spaces is exponentiable in Met if and only if it is exponentiable in PMet and has bounded fibres. Theorem 10. A morphism f : (X, a) → (Y, b) in PMet (Met) is an effective descent map if and only if (∀y0 , y1 , y2 ∈ Y ) b(y2 , y1 ) + b(y1 , y0 ) = inf{a(x2 , x1 ) + a(x1 , x0 ) : xi ∈ f −1 (yi )}. Approach spaces were introduced by R. Lowen [20] as a natural generalisation of both topological and metric spaces. They can be defined in many different

References

375

ways; however, the most convenient presentation for our purpose uses ultrafilter convergence (see also [5]): an approach space (X, a) is a pair consisting of a set X and a numerified convergence structure a : U X ×X → [0, ∞] such that 0 ≥ a(x, ˙ x) and U a(X, x)+a(x, x) ≥ a(mX (X), x). A map f : X → Y between approach spaces (X, a) and (Y, b) is called non-expansive if a(x, x) ≥ b(U f (x), f (x)), for all x ∈ U X and x ∈ X. We denote by App the category of approach spaces and non-expansive maps. So far, in App little is known about exponentiable objects and morphisms and nothing about effective descent morphisms, though one may conjecture that a combination of the known results in Top and Met will provide characterisations of these classes of objects and maps in App. Question 12. Characterise exponentiable objects and maps in App.

845?

Question 13. Characterise effective descent maps in App.

846?

Exponentiable objects in approach theory are studied in [14, 21], and the following sufficient condition is obtained. Theorem 11 ([14]). An approach space (X, a) is exponentiable provided that, for each X ∈ U 2 X, x ∈ X with a(mX (X), x) < ∞, each γ0 , γ1 ∈ [0, ∞) with γ1 + γ0 = a(mX (X), x), and each ε > 0, there exists an ultrafilter x such that U a(X, x) ≤ γ1 + ε and a(x, x) ≤ γ0 + ε. We conjecture that the condition above is also necessary for (X, a) to be exponentiable. A first step towards a solution to Question 13 is to define the functor Conv in the context of approach theory. This, by the way, would also open the door to carry the notion of triquotient map to App. We turn now our attention to the category Unif of uniform spaces and uniformly continuous maps. The question regarding exponentiable maps was settled by S. Niefield [25]. In the result below (X, U) and (Y, V) are uniform spaces, with U and V the sets of entourages of the uniformities. For f : X → Y , A ⊆ X × X and y, y 0 ∈ Y , let Ay,y0 := A ∩ (f −1 (y) × f −1 (y 0 )). Theorem 12. A morphism f : X → Y in Unif is exponentiable if and only if there exists U0 ∈ U satisfying (1) for all U ∈ U, VU = {(y, y 0 ) ∈ Y × Y : U0y,y0 = Uyy0 } ∈ V, (2) there exists V0 ∈ V such that the projection π1 : U0yy0 → f −1 (y) is a surjection whenever (y, y 0 ) ∈ V0 . In particular we have that a uniform space is exponentiable if and only its uniformity has a smallest entourage. However, nothing is known about effective descent maps in Unif . Question 14. Characterise effective descent maps f : X → Y in Unif . References [1] M. Barr, Relational algebras, Reports of the Midwest Category Seminar, IV, Lecture Notes in Math., vol. 137, Springer, Berlin, 1970, pp. 39–55.

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[2] N. Bourbaki, Topologie g´ en´ erale, Hermann & Cie, Paris, 1961. [3] M. M. Clementino, On finite triquotient maps, J. Pure Appl. Algebra 168 (2002), no. 2–3, 387–389. [4] M. M. Clementino and D. Hofmann, Triquotient maps via ultrafilter convergence, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3423–3431. [5] M. M. Clementino and D. Hofmann, Topological features of lax algebras, Appl. Categ. Structures 11 (2003), no. 3, 267–286. [6] M. M. Clementino and D. Hofmann, Effective descent morphisms in categories of lax algebras, Appl. Categ. Structures 12 (2004), no. 5–6, 413–425. [7] M. M. Clementino and D. Hofmann, Exponentiation in V-categories, Topology Appl. 153 (2006), no. 16, 3113–3128. [8] M. M. Clementino, D. Hofmann, and G. Janelidze, Local homeomorphisms via ultrafilter convergence, Proc. Amer. Math. Soc. 133 (2005), no. 3, 917–922. [9] M. M. Clementino, D. Hofmann, and W. Tholen, The convergence approach to exponentiable maps, Port. Math. (N.S.) 60 (2003), no. 2, 139–160. [10] B. Day and G. M. Kelly, On topologically quotient maps preserved by pullbacks or products, Proc. Cambridge Philos. Soc. 67 (1970), 553–558. [11] O. H´ ajek, Notes on quotient maps, Comment. Math. Univ. Carolin. 7 (1966), 319–323. [12] H. Herrlich, E. Lowen-Colebunders, and F. Schwarz, Improving Top: PrTop and PsTop, Category theory at work (Bremen, 1990), Heldermann Verlag, Berlin, 1991, pp. 21–34. [13] D. Hofmann, An algebraic description of regular epimorphisms in topology, J. Pure Appl. Algebra 199 (2005), 71–86. [14] D. Hofmann, Exponentiation for unitary structures, Topology Appl. 153 (2006), no. 16, 3180–3202. [15] G. Janelidze and M. Sobral, Finite preorders and topological descent. I, J. Pure Appl. Algebra 175 (2002), 187–205. ´ [16] G. Janelidze and M. Sobral, Finite preorders and topological descent. II. Etale descent, J. Pure Appl. Algebra 174 (2002), no. 3, 303–309. [17] G. Janelidze, M. Sobral, and W. Tholen, Beyond Barr exactness: effective descent morphisms, Categorical foundations, Cambridge University Press, 2004, pp. 359–405. [18] G. Janelidze and W. Tholen, How algebraic is the change-of-base functor?, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 174–186. [19] G. Janelidze and W. Tholen, Facets of descent. I, Appl. Categ. Structures 2 (1994), no. 3, 245–281. [20] R. Lowen, Approach spaces, The Clarendon Press Oxford University Press, New York, 1997. [21] R. Lowen and M. Sioen, On the multitude of monoidal closed structures on UAP, Topology Appl. 137 (2004), 215–223. [22] E. Lowen-Colebunders and G. Sonck, Exponential objects and Cartesian closedness in the construct Prtop, Appl. Categ. Structures 1 (1993), no. 4, 345–360. [23] E. Michael, Bi-quotient maps and Cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble) 18 (1968), no. 2, 287–302. [24] E. Michael, Complete spaces and tri-quotient maps, Illinois J. Math. 21 (1977), 716–733. [25] S. B. Niefield, Cartesianness: topological spaces, uniform spaces, and affine schemes, J. Pure Appl. Algebra 23 (1982), no. 2, 147–167. [26] T. Plewe, Localic triquotient maps are effective descent maps, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 17–43. [27] J. Reiterman and W. Tholen, Effective descent maps of topological spaces, Topology Appl. 57 (1994), no. 1, 53–69. [28] G. Richter, Exponentiable maps and triquotients in Top, J. Pure Appl. Algebra 168 (2002), no. 1, 99–105. [29] G. Richter, A characterization of exponentiable maps in PrTop, Appl. Categ. Structures 11 (2003), no. 3, 261–265.


Dense subgroups of compact groups W. W. Comfort

1. Introduction The symbol G here denotes the class of all infinite groups, and TG denotes the class of all infinite topological groups which satisfy the T0 separation property. Each element of TG is, then, a Tychonoff space (i.e., a completely regular, Hausdorff space) [50, 8.4]. We say that G = (G, T ) ∈ TG is totally bounded (some authors prefer the expression pre-compact) if for every U ∈ T \ {∅} there is a finite set F ⊆ G such that G = F U . Our point of departure is the following portion of Weil’s Theorem [77]: Every totally bounded G ∈ TG embeds as a dense topological subgroup of a compact group G; this is unique in the sense that e containing G densely there is a homeomorphism-andfor every compact group G e isomorphism ψ : G G fixing G pointwise. As usual, a space is ω-bounded if each of its countable subsets has compact closure. For G = (G, T ) ∈ TG we write G ∈ C [resp., G ∈ Ω; G ∈ CC; G ∈ P; G ∈ TB] if (G, T ) is compact [resp., ω-bounded; countably compact; pseudocompact; totally bounded]. And for X ∈ {C, Ω, CC, P, TB} and G ∈ G we write G ∈ X0 if G admits a group topology T such that (G, T ) ∈ X. The class-theoretic inclusions C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG

and

C ⊆ Ω ⊆ CC ⊆ P ⊆ TB ⊆ TG0 = G 0

0

0

0

0

are easily established. (For P ⊆ TB, see [31]. For TG0 = G, impose on an arbitrary G ∈ G the discrete topology.) We deal here principally with (dense) subgroups of groups G ∈ C, that is, with G ∈ TB. Given G ∈ G we write tb(G) := {T : (G, T ) ∈ TB}. It is good to remember that tb(G) = ∅ and |tb(G)| = 1 are possible (for different G); see 2.3(a) and 5.8(a)(2) below. The symbol A is used as a prefix to indicate an Abelian hypothesis. Thus, for emphasis and clarity: The expression G ∈ AG may be read “G is an infinite Abelian group”, and G ∈ ACC0 may be read “G is an infinite Abelian group which admits a countably compact Hausdorff group topology.” For G, H ∈ G, we write G =alg H to indicate that G and H are algebraically isomorphic. For (Tychonoff) spaces X and Y , we write X =top Y to indicate that the spaces X and Y are homeomorphic. The relation G =alg H promises nothing whatever about the underlying topologies (if any) on G and H; similarly, the relation X =top Y is blind to ambient algebraic considerations (if any). We say that G, H ∈ TG are topologically isomorphic, and we write G ∼ = H, if some bijection between G and H establishes simultaneously both G =alg H and G =top H. 377

§40. Comfort, Dense subgroups of compact groups

378

We distinguish between Problems and Questions. As used here, a Problem is open-ended in flavor, painted with a broad brush; different worthwhile contributions (“solutions”) might lead in different directions. In constrast, a Question here is relatively limited in scope, stated in narrow terms; the language suggests that a “Yes” or “No” answer is desired—although, as we know from experience, that response may vary upon passage from one axiom system to another. With thanks and appreciation I acknowledge helpful comments received on preliminary versions of this paper from: Dikran Dikranjan, Frank Gould, Kenneth Kunen, G´ abor Luk´ acs, Jan van Mill, Dieter Remus, and Javier Trigos-Arrieta.

2. Groups with topologies of pre-assigned type Let X ∈ {C, Ω, CC, P, TB}.

848–852?

Problem 2.1. groups in X0 .

Characterize algebraically the

853–857?

Problem 2.2. Let X ∈ {AC, AΩ, ACC, AP, ATB}. Characterize algebraically the groups in X0 . Discussion 2.3. (a) That ATB0 = AG is easily seen (as in Theorem 3.1(a) below, for example, using the fact that Hom(G, T) separates points of G whenever G ∈ AG). That the inclusion TB0 ⊆ G is proper restates the familiar fact that there are groups G ∈ G whose points are not distinguished by homomorphisms into compact (Hausdorff) groups; in our notation, these are G such that tb(G) = ∅. For example: according to von Neumann and Wigner [59], [50, 22.22(h)], every homomorphism h from the (discrete) special linear group G := SL(2, C) to a compact group satisfies |h[G]| = 1. ˇ (b) It is a consequence of the Cech–Pospisil Theorem [9] (see also [43, Problem 3.12.11], [51, 28.58]) that every G ∈ C satisfies |G| = 2w(G) . Thus in order that G ∈ G satisfy G ∈ C0 it is necessary that |G| have the form |G| = 2κ . (c) The algebraic classification of the groups in AC0 is complete. The full story is given in [50, §25]. (d) It is well known that every G ∈ P0 satisfies |G| ≥ c. See [28] or [13, 6.13] for an explicit proof, and see [8], [42, 1.3] for earlier, more general results. (e) The fact that every pseudocompact space satisfies the conclusion of the Baire Category Theorem has two consequences relating to Problems 2.1 and 2.2. (1) If G ∈ P0 , the cardinal number |G| = κ cannot be a strong limit cardinal with cf(κ) = ω [42]. (2) every torsion group in AP0 is of bounded order [29, 7.4]. (f) No complete characterization of the groups in P0 (nor even in AP0 ) yet exists, but the case of the torsion groups in AP0 is well understood ([26, 40, 41]): A torsion group G ∈ AG of bounded order is in P0 iff for each of its p-primary constituents G(p) each infinite cardinal number of the form κ := |pk ·G(p)| satisfies L 0 and κ is a strong κ Z(p) ∈ P . (Thus for example, as noted in L L[26], if2p is prime Z(p ) ⊕ Z(p) ∈ AP0 while limit cardinal of countable cofinality, then κ 2 κ L L 2 0 / P .) 2κ Z(p) ⊕ κ Z(p ) ∈

Topologies induced by groups of characters

379

(g) An infinite closed subgroup H of G ∈ X ∈ {C, Ω, CC, TB} satisfies H ∈ X, but the comparable assertion for X = P is false [32]. Indeed every H ∈ TB embeds as a closed topological subgroup of a group G ∈ P ([20, 72, 75]). (h) Examples are easily found in ZFC showing that the inclusions AC ⊆ AΩ ⊆ ACC ⊆ AP ⊆ ATB are proper. (See also in this connection 3.3(h) below.) As is indicated in [15, 3.10], the inclusions AC0 ⊆ AΩ0 and ACC0 ⊆ AP0 ⊆ ATB0 are proper in ZFC, but the examples cited there from the literature to show AΩ0 6= ACC0 rest on either CH [70, 71] or MA [73]. Question 2.4. Is there in ZFC a group G ∈ ACC0 \ AΩ0 ? 3. Topologies induced by groups of characters For G ∈ G, we use notation as follows. • H(G) := Hom(G, T), the set of homomorphisms from G to the circle group T. • S(G) is the set of point-separating subgroups of H(G). • When A ∈ S(G), TA is the smallest topology on G with respect to which each element of A is continuous. \ • When (G, T ) ∈ TG, (G, T ) is the set of T -continuous funtions in H(G). These symbols are well-defined for arbitrary groups G, but (in view of the privileged status of the group T) typically they are useful only when G is Abelian. The fact that the groups A ∈ S(G) are required to separate points ensures that the topology TA satisfies the T0 separation property required throughout this article; indeed, the evaluation map eA : G → TA (given by eA (x)h = h(x) for x ∈ G, h ∈ A) is an injective homomorphism, and TA is the topology inherited by G (identified in this context with eA [G]) from TA . When H(G) is given the (compact) topology inherited from TG , a subgroup A ⊆ H(G) satisfies A ∈ S(G) iff A is dense in H(G) (cf. [30, 1.9]). The point of departure for our next problem is this theorem. Theorem 3.1 ([30]). Let G ∈ AG. Then (a) A ∈ S(G) ⇒ (G, TA ) ∈ ATB; (b) (G, T ) ∈ ATB ⇒ ∃ A ∈ S(G) such that T = TA ; \ (c) A ∈ S(G) ⇒ (G, TA ) = A; and (d) A ∈ S(G) ⇒ w(G, TA ) = |A|. It follows from Theorem 3.1(c) that the map A 7→ TA from S(G) to tb(G) is an order-preserving bijection between posets, so tb(G) is large. That theme is noted and developed at length in [4, 5, 25, 27, 30, 34, 63], where the following results (among many others) are given for such G: (a) From |H(G)| = 2|G| ([53], |G| |G| [44, 47.5], [50, 24.47]) and |S(G)| = 22 ([54], [5, 4.3]) follows |tb(G)| = 22 ; |G| |G| (b) from any set of 22 -many elements TA ∈ tb(G), some 22 -many of the spaces (G, TA ) are pairwise nonhomeomorphic; (c) each of the two posets (tb(G), ⊆) and (P(P(|G|)), ⊆) embeds into the other, so any question relating to the existence of a chain or anti-chain or well-ordered set in tb(G) of prescribed cardinality is

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380

independent of the algebraic structure of G and is equivalent to the corresponding strictly set-theoretic question in the poset (P(P(|G|)), ⊆). 859–863?

Problem 3.2. Fix G ∈ AG and fix X ∈ {AC, AΩ, ACC, AP, ATB}. For which A ∈ S(G) is (G, TA ) ∈ X? Discussion 3.3. (a) From 3.1(a), the answer for X = ATB is “all such A”. (b) As mentioned earlier, every group in C0 (whether or not Abelian) satisfies |G| = 2w(G) . Thus for many G ∈ AG (for example, those with cardinality not of the form 2κ ) the answer for X = AC is “no such A”. (c) In parallel with (b), 2.3(e) above indicates that for many G ∈ AG there is no A ∈ S(G) such that (G, TA ) ∈ P. (d) Peripherally or directly, several authors address these questions: Given G ∈ AG, (1) find A ∈ S(G) such that (G, TA ) ∈ AP; or, (2) for which h ∈ H(G) is there A ∈ S(G) such that h ∈ A and (G, TA ) ∈ AP? With no pretense toward completeness I mention in this connection [20, 4.1], [29, 5.11, 6.5], [45], [49, §3, §4], [17, 3.6, 3.10], [57]. (e) For each of the five classes X considered in Problem 3.2, the continuous homomorphic image of each G ∈ X is itself in X. Thus when B ⊆ A ∈ S(G) and B ∈ S(G), from (G, TA ) ∈ X follows (G, TB ) ∈ X. Since a compact (Hausdorff) topology is minimal among Hausdorff topologies, it is immediate that if (G, TA ) ∈ AC and A ⊇ B ∈ S(G), then A = B. (f) That remark leads naturally to less trivial considerations. We say as usual that a group (G, T ) ∈ TG is minimal , and we write (G, T ) ∈ M, if no (T0 ) topological group topology on G is strictly coarser than T . The difficult question, whether (in our notation) AM ⊆ ATB holds, occupied The Bulgarian School for nearly 15 years, finding finally its positive solution in 1984 [61]. (An earlier example had shown that the relation M ⊆ TB is false.) The relevance of the class M to Problem 3.2 is the obvious fact that (G, TA ) ∈ AM iff A is minimal in S(G). We noted in (e) that C ⊆ M, so AC0 ⊆ AM0 , but many G ∈ AG = ATB0 are not in AM0 : the groups Qn (n < ω), Z(p∞ ) are examples. For a careful study, with proofs and historical commentary and a comprehensive bibliography, of the groups which are/are not in the classes M, AM, M0 , AM0 , including a proof of the theorem AM ⊆ ATB, the reader should consult [38]. See also [36] for background on the principal remaining outstanding problem in this area: Which reduced, torsion-free G ∈ AG are in AM0 ? b is uniformly (g) Let D be a dense subgroup of E ∈ ATG. Since each h ∈ D b we continuous and T is complete, each such h extends (uniquely) to h ∈ E; Q b b have, then, D =alg E. Now let {Gi : i ∈ I} ⊆ ATB and set G := i∈I Gi . Q From the uniqueness aspect of Weil’s theorem we have G = i∈I Gi ; further, the c is well known. (Here L G c is given the discrete topology. b =L G relation G i∈I

i

i∈I

i

See [50, 23.21] for a comprehensive treatment.) It follows from the Tychonoff Product Theorem that if each Gi ∈ X ∈ {C, Ω, TB} then also G ∈ X; it is shown in [30] that, similarly, if each Gi ∈ P then also G ∈ P.

Topologies induced by groups of characters

381

The relevance of the foregoing paragraph to Problem 3.2 is this: If X ∈ {C, Ω, P, TB} L and {Gi : i ∈ I} ⊆ AG, and if Ai ∈ S(Gi ) makes (Gi , TAi ) ∈ X, then A := i∈I Ai makes also (G, TA ) ∈ X. This indicates a global coherence or stability in passing among the groups G ∈ AG while seeking A ∈ S(G) for which (G, TA ) ∈ X. [Remark. In contrast to the classes C, Ω, P and TB, the issue of “the closure under the formation of products” in the class CC is much more complex and subtle. See [39] for an extended discussion and many relevant problems; see also Question 5.1 below.] (h) The algebraic structure of a group A ∈ S(G) is not of itself sufficient to determine whether or not (G, TA ) ∈ X. For examples to thatL effect, begin with K := {0, 1}c in its usual compact topology and define G := c {0, 1}. (0) Let G0 be the Σ-product in K; (1) Let D be a countable, dense subgroup of K and let G1 ∈ CC satisfy |G1 | ≤ c and D ⊆ G1 ⊆ K. [Such a group G1 may be defined by choosing for each countable set C ⊆ D an accumulation point pC of C in K, forming the subgroup of K generated by D and all points pC , and iterating the process over the countable ordinals; see [32] for details, and see [65] for the modified argument furnishing such G1 which is p-compact (in the sense of 5(A) below) for pre-assigned p ∈ ω ∗ .] (2) Let G2 be a proper, dense pseudocompact subgroup of G0 . [Such a group is given in [29, 7.3], [21].] (3) Write K = K0 × K1 with Ki ∼ = K, let H and E be the natural copies of G0 and D in K0 and K1 , respectively, and set G3 := H × E ⊆ K. Then each Gi is dense in K, with |Gi | ≤ c by construction. For 0 ≤ i ≤ 2 we have |Gi | ≥ c by 2.3(d), and also |G3 | = |G0 | · |D| ≥ c. Thus each Gi is a Boolean group with |Gi | = c, so Gi =alg G for 0 ≤ i ≤ 3 [50, A.25]. Let Ti be the topology on Gi inherited from K, and (using Theorem 3.1(b) above) let Ai ∈ S(G) satisfy (G, TAi ) ∼ = (Gi , Ti ). Again, since Ai is a Boolean group with |Ai | = w(G, TAi ) = c, we have Ai =alg ⊕c {0, 1} for 0 ≤ i ≤ 3. Then we have (G, TA0 ) ∼ = G0 ∈ Ω \ C. (Proof. |G0 | = c < 2c = |K|.) (G, TA1 ) ∼ = G1 ∈ CC \ Ω. (Proof. D is dense in G1 with |D| = ω, so G1 ∈ Ω gives the contradiction G1 = K.) (G, TA2 ) ∼ = G2 ∈ P \ CC. (Proof. The general result given in [12], [29, 3.3] shows that no proper, Gδ -dense subset of G0 is countably compact.) (G, TA3 ) ∼ = G3 ∈ TB \ P. (Proof. If G3 ∈ P then also its continuous image E = π1 [G3 ] ∈ P, which with |E| = ω < c contradicts 2.3(d).) Without specific reference to any of the classes X here considered, two questions now arise naturally. (See also Problem 5.6 below for a related query.) Problem 3.4. Fix G ∈ AG. For which A, B ∈ S(G) does the relation (G, TA ) ∼ = (G, TB ) hold? Problem 3.5. Fix G ∈ AG. For which A, B ∈ S(G) does the relation (G, TA ) =top (G, TB ) hold? Discussion 3.6. (a) A simple example, easily generalized, will suffice to indicate a distinction between Problems 3.4 and 3.5. Choose A, B ∈ S(Z) such that |A| = |B| = ω and A 6=alg B. The spaces (Z, TA ), (Z, TB ) are then countable

864?

865?

382


and metrizable without isolated points, hence according to a familiar theorem of Sierpi´ nski [66] (see also [43, Exercise 6.2.A(d)]) are homeomorphic. So here (Z, TA ) =top (Z, TB ) and (Z, TA ) =alg (Z, TB ), but (Z, TA ) ∼ = (Z, TB ) fails since \ \ from Theorem 3.1(c) we have (Z, TA ) = A 6=alg B = (Z, TB ). (b) It is evident from Theorem 3.1(d) that if (G, TA ) =top (G, TB ) then |A| = |B|. The converse fails, however, even in the case G = Z: It L is shown in [62] that there is a set {Aη : η < c} ⊆ S(Z), with each Aη =alg c Z, such that (Z, TAη ) 6=top (Z, TAη0 ) for η < η 0 < c; one may arrange further that all, or none, of the spaces (Z, Tη ) contain a nontrivial convergent sequence. Similar results in a more general context are given in [24]. (c) In groups of the form H(G) ∈ AC with G ∈ AG, Dikranjan [37] has introduced a strong density concept, called g-density, which is enjoyed by certain A ∈ S(G). We omit the formal definition here, but we note that A ∈ S(G) is g-dense in H(G) iff no nontrivial sequence converges in (G, TA ) [37, 4.22]. Thus for certain pairs A, B ∈ S(G), g-density successfully responds to Problem 3.5: If (G, TA ) =top (G, TB ), then A is g-dense in H(G) iff B is g-dense in H(G). For additional theorems on g-dense and g-closed subgroups of the groups H(G) ∈ AC, see [3, 56]. 4. Extremal phenomena It is obvious that a group (G, T ) ∈ C admits neither a proper, dense subgroup in C nor a strictly larger group topology U such that (G, U) ∈ C. A brief additional argument ([33, 3.1], [29, 3.2]) shows that if (G, T ) ∈ P is metrizable (equivalently: satisfies w(G, T ) = ω), then (1) (G, T ) ∈ C and (2) (G, T ) admits neither a proper, dense subgroup in P nor a strictly larger group topology U such that (G, U) ∈ P. Indeed, for (G, T ) ∈ AP these conditions are equivalent [21]: (a) (G, T ) admits a proper, dense subgroup in P; (b) there is a topology U on G, strictly refining T , such that (G, U) ∈ P; (c) w(G, T ) > ω. Some questions then arise naturally. 866?

Question 4.1. Do the three conditions just listed from [21] remain equivalent for (G, T ) ∈ P when G is not assumed to be Abelian? When G ∈ AG is given the topology TA with A = H(G), every subgroup of G is closed; so, (G, TA ) ∈ ATG has no proper dense subgroup. As to the existence of strict refinements, it is clear for every G ∈ TB0 that tb(G) has a largest (= biggest) topology. (When G ∈ AG ⊆ TB0 this is T = TA with A = H(G).) The class X = TB, then, is included in this next Problem largely to preserve the symmetry developed in this essay; in this immediate context it deserves minimal attention. As to Problem 4.3, we leave it to the reader to determine which pairs X, Y give rise to questions of interest and which to questions which are silly or without content.

867–868?

Problem 4.2. Fix X ∈ {Ω, CC, TB}, and let (G, T ) ∈ X. Find pleasing necessary and/or sufficient conditions that (a) (G, T ) has a proper, dense subgroup in X; and/or (b) there is a topology U on G, strictly refining T , such that (G, U) ∈ X.

Miscellaneous questions

383

Or even, more generally: Problem 4.3 ([15]). Fix X, Y ∈ {C, Ω, CC, P, TB}, and let (G, T ) ∈ X. Find pleasing necessary and/or sufficient conditions that (a) (G, T ) has a proper, dense subgroup in Y; and/or (b) there is a topology U on G, strictly refining T , such that (G, U) ∈ Y.

869–870?

Remark 4.4. We noted already in 3.3(h) that according to [12], [29, 3.3] the Σ-product in a group of the form Gκ with G ∈ C and κ ≥ ω admits no proper dense subgroup in CC. 5. Miscellaneous questions (A). Recall first a definition of A.R. Bernstein [6]: For an ultrafilter p ∈ ω ∗ := β(ω) \ ω, a Tychonoff space X is p-compact if for every (continuous) f : ω → X ˇ the Stone–Cech extension f : β(ω) → β(X) satisfies f (p) ∈ X. (See [13, 14, 65] for references to related tools introduced by Frol´ık, by Katˇetov, and by V. Saks.) It is known [11, 47,Q65] that for a set {Xi : i ∈ I} of Tychonoff spaces, every product of the form i∈I (Xi )κi is countably compact iff there is p ∈ ω ∗ such that each Xi is p-compact. It is then immediate, as in [13, 8.9], that the class CC is closed under the formation of (arbitrary) products if and only if there is p ∈ ω ∗ such that each G ∈ CC is p-compact. Thus we are led to a bizarre question. Question 5.1 ([14, 1.A.1]). Is it consistent with the axioms of ZFC that there is p ∈ ω ∗ such that every countably compact group is p-compact?

871?

Discussion 5.2. The chapter [39] makes reference to work of E. van Douwen, J. van Mill, K.P. Hart, A. Tomita and others giving the existence of models of ZFC in which some product of finitely many elements from the class ACC fails to be in ACC. See also [46] for a similar conclusion based on the existence of a selective ultrafilter p ∈ ω ∗ . It is evident from (A) above that in any of these models of ZFC no ultrafilter p as in Question 5.1 can exist. (B). It was noted in 3.3(g) that when D is a dense subgroup of G ∈ ATG, bD b given by φ(h) = h|D establishes the equality G b =alg D; b clearly the map φ : G b b φ is continuous when D and G are given their respective compact-open topologies. Following [22, 23], we say that G ∈ ATG is determined if for each dense subgroup bD b is a homeomorphism. The principal theorem in this D of G the map φ : G area, given in [10] and [1] independently, is this: Every metrizable G ∈ ATG is determined. (The exact generalization of that theorem to the (possibly) nonAbelian context is given in [55]: For every dense subgroup D of metrizable G ∈ TG and for every compact Lie group K, one has Hom(D, K) ∼ = Hom(G, K) when those groups are given the compact-open topology.) The authors of [22, 23] have noted the existence of many nonmetrizable determined G ∈ ATG (some of them compact), and they raised this question (see also [18, §6]). Question 5.3. Is the product of finitely many determined groups in ATG necessarily determined? If G ∈ ATG is determined, must G × G be determined?

872–873?

384


Discussion 5.4. Concerning Question 5.3, it is known [74] that the product of two determined groups in ATG, of which one is discrete, is again determined. Now let non(N ) be the least cardinal κ such that some set X ⊆ T with |X| = κ has positive outer (Haar) measure. It is known [22, 23] that no G ∈ AC with w(G) ≥ non(N ) is determined, so if non(N ) = ℵ1 (in particular, if CH holds) then G ∈ AC is determined iff w(G) = ω (i.e., iff G is metrizable). The authors of [22, 23] asked Question 5.5, a sharpened version of [18, 6.1]. 874–876?

Question 5.5. Is there in ZFC a cardinal κ such that G ∈ AC is determined iff w(G) < κ? Is κ = non(N )? Is κ = ℵ1 ? (C). The remark in 3.6(a) shows for G, H ∈ ATG that the conditions G =alg H, G =top H do not together ensure that G and H are topologically isomorphic, b and H, b respectively. The even when G and H carry the topologies induced by G following problem then arises naturally.

877–878?

Problem 5.6. (a) Find interesting necessary and/or sufficient conditions on G, H ∈ TG to ensure that if G =alg H and G =top H then necessarily G ∼ = H. (b) Find interesting necessary and/or sufficient conditions on G, H ∈ ATG to ensure that if G =alg H and G =top H then necessarily G ∼ = H. Problem 5.6 relates to pairs from TG. A similar problem focuses on a fixed G ∈ TG, as follows.

879–880?

Problem 5.7. (a) For which G ∈ TG do the conditions H ∈ TG, G =alg H and G =top H guarantee that G ∼ = H? (b) For which G ∈ ATG do the conditions H ∈ ATG, G =alg H and G =top H guarantee that G ∼ = H? Discussion 5.8. (a) There are many theorems in the literature showing that certain G ∈ G admit a topology with certain pre-assigned properties, further that any two such topologies T0 , T1 satisfy (G, T0 ) =top (G, T1 ), or (G, T0 ) ∼ = (G, T1 ), or even T0 = T1 . [Remark. In this last case, every automorphism of G is a T0 homeomorphism.] We cite six results of this and similar flavor; clearly, these relate closely to the issues raised in Problems 5.6 and 5.7. (1) Van der Waerden [76] gave examples of groups (G, T ) ∈ C such that tb(G) = {T }. (2) Groups as in (1) are necessarily metrizable [48, 58], but there exist G ∈ G of arbitrary cardinality ≥ c with |tb(G)| = 1: [27, 3.17] shows that for every family {Gi : i ∈ I} ⊆ C of algebraically simple, non-Abelian Lie groups, the only topology T on the group L H := i∈I Gi for which Q (H, T ) ∈ TB is the topology inherited from the (usual compact) topology on i∈I Gi . See also [64] for additional relevant references. (3) [Hulanicki, Orsatti] On an Orsatti group—i.e., a group G of the form G =alg Q kp p∈P (Zp × Fp ) with Zp the p-adic integers and with finite Fp ∈ AG such that p · Fp = {0}—the obvious natural topology T making (G, T ) ∈ C is the only topology making G ∈ C; conversely, every G ∈ AG with a unique topology T making (G, T ) ∈ C is an Orsatti group. (4) [Stewart] If G ∈ G admits a connected topology T such that (G, T ) ∈ C and the center of G is totally disconnected, then T is the only topology making (G, T ) ∈ C. (5) [Scheinberg] If G, H ∈ ATG are

Miscellaneous questions

385

connected and (locally) compact and G =top H, then G ∼ = H. (6) Every totally disconnected G ∈ C with w(G) = κ satisfies G =top {0, 1}κ ; if further there is p ∈ P such that p · G = {0}, then G ∼ = (Z(p))κ . (b) We note en passant that in a model Lof ZFC with distinct cardinals κi (i = 0, 1) such that 2κ0 = 2κ1 , the group G = 2κi {0, 1} =alg {0, 1}κi admits totally disconnected (compact) topologies Ti such that w(G, Ti ) = κi , hence (G, T0 ) 6=top (G, T1 ). (c) For a detailed discussion of the results cited in (a)(3)–(a)(6), with references to works of Stewart, Hulanicki, Scheinberg, Orsatti and others, see [19, 35, 38, 50, 52, 64]. (d) Suppose for some G ∈ G and for one of the classes X ∈ {C, Ω, CC, P} that (1) G ∈ X0 and (2) every two topologies T0 , T1 making (G, Ti ) ∈ X satisfy (G, T0 ) ∼ = (G, T1 ). Then G = (G, T0 ) is an example of the sort sought in Problem 5.7. [Proof. From (G, T0 ) ∈ X and G =top H ∈ TG follows H ∈ X, and any (hypothesized) isomorphism φ : H G induces on G a topology T1 such that φ is a homeomorphism and (G, T1 ) ∈ X. Then (G, T0 ) ∼ = (G, T1 ) ∼ = H.] (D). A subgroup G of K ∈ TG is said to be essentially dense [resp., totally dense] in K if |G ∩ N | > 1 [resp., G ∩ N is dense in N ] for every closed, normal, nontrivial subgroup N of G. Given K ∈ TG, the essential density ed(K) [resp., the total density td(K)] of K is the cardinal number ed(K) := min{|G| : G is an essentially dense subgroup of K}, td(K) := min{|G| : G is a totally dense subgroup of K}. In contrast with the properties studied heretofore in this article—compactness, pseudocompactness, and so forth—the properties of essential and total density are not intrinsic to a group G ∈ TG: They must be investigated relative to an enveloping group K ∈ TG. It is known [2, 60, 67] for G dense in K ∈ TG that G ∈ M iff K ∈ M and G is essentially dense in K. Hence, since C ⊆ M, for G ∈ AG these properties are equivalent: (1) (G, TA ) ∈ AM; (2) (G, TA ) is essentially dense in (G, TA ); (3) A is minimal in S(G). We are drawn to the companion problem for total density. Problem 5.9. For which A ∈ S(G) is (G, TA ) totally dense in (G, TA )?

881?

It is shown in [16] that there are G ∈ ATB such that ed(G) < td(G), and that consistently such G ∈ AP exist. The authors of [16] leave several related questions unanswered, however, including these. Question 5.10 ([16]). (a) Is there, in ZFC or in augmented axiom systems, a group G ∈ ACC such that ed(G) < td(G)? (b) Is there in ZFC a group G ∈ AP such that ed(G) < td(G)? Discussion 5.11. Every G ∈ AC satisfies ed(G) = td(G) [69]. The paper [7] provides much additional useful background for 5.9 and 5.10.

882–883?

386


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Selected topics from the structure theory of topological groups Dikran Dikranjan and Dmitri Shakhmatov This article contains open problems and questions covering the following topics: the dimension theory of topological groups, pseudocompact and countably compact group topologies on Abelian groups, with or without nontrivial convergent sequences, categorically compact groups, sequentially complete groups, the Markov–Zariski topology, the Bohr topology, and transversal group topologies. All topological groups considered in this chapter are assumed to be Hausdorff. 1. Dimension theory of topological groups We highlight here our favourite problems from the dimension theory of topological groups. Problem 1 ([1]). Is ind G = Ind G = dim G for a topological group G with a countable network?

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The classical result of Pasynkov says that ind G = Ind G = dim G for a (locally) compact group G [50]. Question 2 ([52]). Is ind G = Ind G = dim G for a σ-compact group G?

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This is a delicate question since there exists an example of a precompact topological group G such that G is a Lindelöf Σ-space, dim G = 1 but ind G = Ind G = ∞ [52, 53]. Even the following particular case of Question 2 seems to be open. Question 3 (M.J. Chasco). If a topological group G is a kω -space, must ind G = Ind G = dim G?

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Recall that X is a kω -space provided that there exists a countable family {Kn : n ∈ ω} of compact subspaces of X such that a subset U of X is open in X if and only if U ∩ Kn is open in Kn for every n ∈ ω. Question 4 ([53]). Is ind G = Ind G for a Lindel¨ of group G? The answer to Question 4 is positive if G is a Lindelöf Σ-space (in particular, a σ-compact space), so only the inequality ind G ≤ dim G must be proved in order to answer Questions 2 or 3 positively. The first named author was partially supported by the project MIUR 2005 “Anelli commutativi e loro moduli: teoria moltiplicativa degli ideali, metodi omologici e topologici.” The second named author acknowledges partial financial support from the Grant-in-Aid for Scientific Research no. 155400823 by the Japan Society for the Promotion of Science. 389

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§41. Dikranjan and Shakhmatov, Structure theory of topological groups

Problem 5 (Old problem). If H is a subgroup of a topological group G, is then dim H ≤ dim G? The answer is positive if H is R-factorizable [60] (in particular, precompact [54]).

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Question 6 ([55]). Suppose that X is a separable metric space with dim X ≤ n. Is there a separable metric group G that contains X as a closed subspace such that dim G ≤ 2n + 1?1 Without the requirement that X is closed in G the answer is positive due to the N¨ oebeling–Pontryagin theorem: X is a subspace of the topological group R2n+1 . The separability in the above question is essential: There exists a metric space X of weight ω1 such that dim X = 1 and X cannot be embedded into any finite-dimensional topological group [42]. The next question is the natural group analogue of the classical result about the existence of the universal space of a given weight and covering dimension.

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Question 7 ([55]). Let τ be an infinite cardinal and n be a natural number. Is there an (Abelian) topological group Hτ,n with dim Hτ,n ≤ n and w(Hτ,n ) ≤ τ such that every (Abelian) topological group G satisfying dim G ≤ n and w(G) ≤ τ is topologically and algebraically isomorphic to a subgroup of Hτ,n ? The special case of the above question when τ = ω is due to Arhangel0 ski˘ı [1]. Transfinite inductive dimensions have many peculiar properties in topological groups [56]. For example, (i) if G is a locally compact group having small transfinite inductive dimension trind(G), then G must be finite-dimensional, and (ii) if G is a separable metric group having large transfinite inductive dimension trInd(G), then G must be finite-dimensional as well. It is not clear if (ii) holds for trind(G) instead of trInd(G):

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Problem 8 ([56]). For which ordinals α does there exist a separable metric group Gα whose small transfinite inductive dimension trind(Gα ) equals α? The reader is referred to [41, 55, 56] for additional open problems in the dimension theory of topological groups. 2. Pseudocompact and countably compact group topologies on Abelian groups We denote by C the class of Abelian groups that admit a countably compact group topology, and use P to denote the class of Abelian groups that admit a pseudocompact group topology. The next two problems are the most fundamental problems in this area: 1We were kindly informed by T. Banakh that Question 6 has been recently answered in the negative in [2]: There exists a 1-dimensional separable metric space (namely, the hedgehog with countably many spines) which cannot be embedded into any finite-dimensional topological group as a closed subspace.

Pseudocompact and countably compact group topologies on Abelian groups

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Problem 9 ([19]). Describe the algebraic structure of members of the class P.

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Problem 10. Describe the algebraic structure of members of the class C.

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Despite a substantial progress on Problem 9 for particular classes of groups achieved in [7, 8, 16–18], the general case is still very far from the final solution. (We refer the reader to [4] for further reading on this topic.) Let G be an Abelian group. As usual r(G) denotes the free rank of G. For every natural number n ≥ 1 define G[n] = {g ∈ G : ng = 0} and nG = {ng : g ∈ G}. Every group G from the class C satisfies the following two conditions [16, 18, 26]: PS: Either r(G) ≥ c or G = G[n] for some n ∈ ω \ {0}. CC: For every pair of integers n ≥ 1 and m ≥ 1 the group mG[n] is either finite or has size at least c. It is totally unclear if these are the only necessary conditions on a group from the class C: Question 11. Is it true that an Abelian group G belongs to C if and only if G satisfies both PS and CC? Question 12 ([22]). Is it true in ZFC that an Abelian group G of size at most 2c belongs to C if and only if G satisfies both PS and CC?

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Question 12 has a positive consistent answer [22]. Assuming MA, there exist countably compact Abelian groups G, H such that G × H is not countably compact [31]. Therefore, our next question could be viewed as a weaker form of productivity of countable compactness in topological groups that still has a chance for a positive answer in ZFC. Question 13 ([19]). If G and H belong to C, must then their product G × H also belong to C?

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In fact, one can consider a much bolder hypothesis: Question 14 ([19]). Is C closed Q under arbitrary products? That is, if Gi belongs to C for each i ∈ I, does then i∈I Gi belong to C?

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The next question provides a slightly less bold conjecture: Question Q 15 ([19]). (i) Is there a cardinal Q τ having the following property: A product i∈I Gi belongs to C provided that j∈J Gj belongs to C whenever J ⊆ I and |J| ≤ τ ? (ii) Does the statement in item (i) hold true when τ = c or τ = 2c ? Of course Question 14 simply asks if the statement in item (i) of Question 15 holds true when τ = 1. It might be worth noting that Q Question 15 is motivated by a theorem of Ginsburg and Saks [35]:Q A product i∈I Xi of topological spaces Xi is countably compact provided that j∈J Gj is countably compact whenever J ⊆ I and |J| ≤ 2c . A partial positive answer to Question 14 has been given in [20]: It is consistent with ZFC that, for every family {Gi : i ∈ I} of groups with 2|I| ≤ 2c such that

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Q Gi belongs to C and |Gi | ≤ 2c for each i ∈ I, the product i∈I Gi also belongs to C. A similar result for smaller products and smaller groups has been proved in [26, Theorem 5.6] under the assumption of MA. In particular, if the groups G and H in Question 13 are additionally assumed to be of size at most 2c , then the positive answer to this restricted version of Question 13 is consistent with ZFC [20]. Recall that an Abelian group G is algebraically compact provided that one can find an Abelian group H such that G × H admits a compact group topology. Algebraically compact groups form a relatively narrow subclass of Abelian groups (for example, the group Z of integers is not algebraically compact). On the other hand, every Abelian group G is algebraically pseudocompact; that is, one can find an Abelian group H such that G × H ∈ P [18, Theorem 8.15]. It is unclear if this result can be strengthened to show that every Abelian group is algebraically countably compact: 900?

Question 16 ([22]). Given an Abelian group G, can one always find an Abelian group H such that G × H ∈ C? Recall that an Abelian group G is divisible provided that for every g ∈ G and each positive integer n one can find h ∈ G such that nh = g. An Abelian group is reduced if it does not have non-zero divisible subgroups. Every Abelian group G admits a unique representation G = D(G) × R(G) into the maximal divisible subgroup D(G) of G (the so-called divisible part of G) and the reduced subgroup R(G) ∼ = G/D(G) of G (the so-called reduced part of G). It is well-known that an Abelian group G admits a compact group topology if and only if both its divisible part D(G) and its reduced part R(G) admit a compact group topology. However, there exist groups G and H that belong to P but neither D(G) nor R(H) belong to P [18, Theorem 8.1(ii)]. This was strengthened in [22, 26] as follows: It is consistent with ZFC that there exist groups G0 and H 0 from the class C such that neither D(G0 ) nor R(H 0 ) belong to P. These results leave open the following:

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Problem 17 ([19]). In ZFC, give an example of groups G and H from the class C such that: (i) D(G) does not belong to C (or even P), (ii) R(H) does not belong to C (or even P). Even the following question is also open:

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Question 18 ([19]). Let G be a group in C. (i) Is it true that either D(G) or R(G) belongs to C? (ii) Must either D(G) or R(G) belong to P? We note that item (ii) of the last question is a strengthening of Question 9.8 from [18]. Even consistent results related to the last question are currently unavailable. S An Abelian group G is torsion S provided that G = {G[n] : n ∈ ω, n ≥ 1}, and is torsion-free provided that {G[n] : n ∈ ω, n ≥ 1} = {0}.

Properties determined by convergent sequences

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Question 19 ([22]). Is there a torsion Abelian group that is in P but not in C?

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Question 20 ([22]). Is there a torsion-free Abelian group that is in P but not in C?

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It is consistent with ZFC that a group Questions 19 and 20 must have size strictly bigger than 2c [22]. Problem 21 ([22]). (i) Describe in ZFC the algebraic structure of separable countably compact Abelian groups. (ii) Is it true in ZFC that an Abelian group G admits a separable countably compact group topology if and only if |G| ≤ 2c and G satisfies both PS and CC?

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A consistent positive solution to Problem 21(ii) is given in [22]. 3. Properties determined by convergent sequences It is well-known that infinite compact groups have (lots of) nontrivial convergent sequences. There exists an example (in ZFC) of a pseudocompact Abelian group without nontrivial convergent sequences [58]. While there are plenty of consistent examples of countably compact groups without nontrivial convergent sequences [10, 22, 26, 31, 39, 46, 59, 63], the following remains a major open problem in this area: Problem 22. Does there exist, in ZFC, a countably compact group without nontrivial convergent sequences?

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Recall that a (Hausdorff) topological group G is minimal if G does not admit a strictly weaker (Hausdorff) group topology. Even though a countably compact, minimal Abelian group need not be compact, it can be shown that it must contain a nontrivial convergent sequence. More generally, one can show that an infinite, countably compact, minimal nilpotent group has a nontrivial convergent sequence. Whether “nilpotent” can be dropped remains unclear. Problem 23. Must an infinite, countably compact, minimal group contain a nontrivial convergent sequence?

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The next question may be considered as a countably compact (or pseudocompact) heir of the fact that compact groups have nontrivial convergent sequences that still has a chance of a positive answer in ZFC. Question 24 ([22]). Let G be an infinite group admitting a countably compact (or a pseudocompact) group topology. Does G have a countably compact (respectively, pseudocompact) group topology that contains a nontrivial convergent sequence?

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The next question goes in the opposite direction: Question 25 ([22]). (i) Does every group G admitting a pseudocompact group topology have also a pseudocompact group topology without nontrivial convergent sequences (without infinite compact subsets)?

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(ii) Does every group G admitting a countably compact group topology have also a countably compact group topology without nontrivial convergent sequences (without infinite compact subsets)? Question 25(ii) has a consistent positive answer in the special case when |G| ≤ 2c [22]. The part “without nontrivial convergent sequences” of item (ii) of our next question has appeared in [10]. 914–916?

Question 26. (i) When does a compact Abelian group G admit a proper dense subgroup H without nontrivial convergent sequences? without infinite compact subsets? (ii) When does a compact Abelian group G admit a proper dense pseudocompact subgroup H without nontrivial convergent sequences? without infinite compact subsets? (iii) When does a compact Abelian group G admit a proper dense countably compact subgroup H without nontrivial convergent sequences? without infinite compact subsets? In GCH, a precompact group H such that w(H) < w(H)ω has a nontrivial convergent sequence [47]. Thus w(G) = w(G)ω is a necessary condition for the group G to have aQsubgroup H as in Question 26. This condition alone is not sufficient: If K = n∈ω Z2n and τ is an infinite cardinal, then every dense subgroup H of G = K τ has a nontrivial convergent sequence [10] (here Zm denotes the cyclic group Z/mZ). Many partial results towards solution of Question 26 are given in [10, 33].

917–918?

Question 27. (i) If a compact Abelian group has a proper dense pseudocompact subgroup without nontrivial convergent sequences, does it also have a proper dense pseudocompact subgroup without infinite compact subsets? (ii) If a compact Abelian group has a proper dense countably compact subgroup without nontrivial convergent sequences, does it also have a proper dense countably compact subgroup without infinite compact subsets? Now we relax item (ii) to get the following:

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Question 28. Is the existence of a countably compact Abelian group without nontrivial convergent sequences equivalent to the existence of a countably compact Abelian group without infinite compact subsets? In connection with the last four questions we should note that, under MA, an infinite compact space of size at most c contains a nontrivial convergent sequence. A topological group G is called sequentially complete [24, 25] if G is sequentially closed in every (Hausdorff) group that contains G as a topological subgroup. Obviously, every topological group without nontrivial convergent sequences is sequentially complete. Moreover, sequential completeness is preserved under taking arbitrary direct products and closed subgroups [24]. Denote by S the class of closed subgroups of the products of countably compact Abelian groups. Since countably compact groups are sequentially complete and precompact, every group from the class S is sequentially complete and precompact.

Categorically compact groups

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Question 29 ([25]). (i) Does every precompact sequentially complete Abelian group G belong to S? (ii) What is the answer to (i) if one additionally assumes that |G| ≤ c?

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Every precompact Abelian group is both a quotient group and a continuous isomorphic image of some sequentially complete precompact Abelian group [25, Theorem B]. This motivates the following: Question 30 ([25]). (i) a quotient of (ii) a continuous (iii) a continuous

Is every precompact Abelian group G: a group from S? homomoprhic image of group from S? isomorphic image of group from S?

922–924?

Item (iii) of Question 30 has a positive answer when |G| ≤ c [25, Theorem A], and more generally, if |G| is a non-measurable cardinal [61]. 4. Categorically compact groups A topological group G is categorically compact (briefly, c-compact) if for each topological group H the projection G × H → H sends closed subgroups of G × H to closed subgroups of H [29]. Obviously, compact groups are c-compact. To establish the converse is the main open problem in this area: Problem 31. (i) Are c-compact groups compact? (ii) Are nondiscrete c-compact groups compact?

925–926?

Item (i) has appeared in [29]. Two related weaker versions are also open: Question 32. Is every (nondiscrete) c-compact group minimal?

927?

Question 33. Does every nondiscrete c-compact group have a nontrivial convergent sequence?

928?

A positive answer to Problem 31 in the Abelian case makes recourse to the deep theorem of precompactness of Prodanov and Stoyanov [15]. Similar to (usual) compactness, taking products, closed subgroups and continuous homomorphic images preserves c-compactness [29] (a proof of the productivity of c-compactness was obtained independently also in [3] in a much more general setting). Therefore, a positive answer to Question 32 would imply that every closed subgroup H of a c-compact group is totally minimal , i.e., all quotient groups of H are minimal. At present we only know that separable c-compact groups are totally minimal (and complete) [29]. Lukacs [45] resolved Problem 31 positively for maximally almost periodic groups. Moreover, he showed that it suffices to solve this problem only for second countable groups (analogously, the case of locally compact SIN-groups, is reduced to that of countable discrete groups [45]). (Recall that a SIN-group is a topological group for which the left and right uniformities coincide.) According to [45], in Question 33 it suffices to consider only the nondiscrete c-compact groups that have no nontrivial continuous homomorphisms into compact groups.

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Connected locally compact c-compact groups are compact [29]. Hence the connected locally compact group SL2 (R) is not categorically compact, although it is separable and totally minimal [15]. Nothing is known about c-compactness of disconnected locally compact groups. In fact, even the discrete case is wide open: 929?

Question 34 ([29]). Is every discrete c-compact group finite (finitely generated, of finite exponent, countable)? One can prove that a countable discrete group G is c-compact if and only if every subgroup of G is totally minimal [29]. Therefore, the negative answer to this question is equivalent to the existence of an infinite group G such that no subgroup or quotient group of G admits a nondiscrete Hausdorff group topology (this is a stronger version of the famous Markov problem on the existence of a countably infinite group without nondiscrete Hausdorff group topologies). A group G is h-complete if all continuous homomorphic images of G are complete, and G is hereditarily h-complete if every closed subgroup of G is h-complete. c-compact groups are hereditarily h-complete, and the inverse implication holds for SIN-groups (in particular, Abelian groups) [29]. Both c-compactness and h-completeness are stable under products, and hcompleteness also has the the so-called three space property: If K is a closed normal subgroup of a topological group G such that both K and the quotient group G/K are h-complete, then G is h-complete. This leaves open:

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Question 35 ([29, Question 4.3]). If K is a closed normal subgroup of a topological group G such that both K and the quotient group G/K are c-compact, must G be c-compact as well? Nilpotent (in particular, Abelian) h-complete groups are compact, while solvable c-compact groups are compact [29]. This motivates the following:

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Question 36 ([29, Question 3.13]). Are solvable h-complete groups c-compact? 5. The Markov–Zariski topology According to Markov [48], a subset S of a group G is called: (a) elementary algebraic if there exist an integer n > 0, a1 , . . . , an ∈ G and ε1 , . . . , εn ∈ {−1, 1} such that S = {x ∈ G : xε1 a1 xε2 a2 . . . an−1 xεn = an }, (b) algebraic if S is an intersection of finite unions of elementary algebraic subsets, (c) unconditionally closed if S is closed in every Hausdorff group topology of G, (d) potentially dense if G admits a Hausdorff group topology T on G such that S is dense in (G, T ). The family of algebraic subsets of a group G coincides with the family of closed subsets of a T1 topology ZG on G, called the Zariski topology. The family of unconditionally closed subsets of G coincides with the family of closed subsets of a

The Markov–Zariski topology

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T1 topology MG on G, namely the infimum (taken in the lattice of all topologies on G) of all Hausdorff group topologies on G. We call MG the Markov topology of G. Analogously, let PG be the infimum of all precompact Hausdorff group topologies on G (if G admits no such topologies let PG denote the discrete topology of G). It seems natural to call PG the precompact Markov topology of G. Note that (G, ZG ), (G, MG ) and (G, PG ) are quasi-topological groups, i.e., the inversion and shifts are continuous. Clearly, ZG ⊆ MG ⊆ PG . If G is Abelian, then ZG = MG = PG [21]. Markov has attributed the equality ZG = MG in the Abelian case to Perel0 man but the proof has never appeared in print. Another proof was recently announced by Sipacheva [41]. In the particular case when G is almost torsion-free2 the equality ZG = MG was earlier proved in [62]. Problem 37 ([48]). Does ZG = MG hold true for an arbitrary group G?

932?

The answer is positive when G is countable [48]. A consistent counterexample to this question was announced quite recently by Sipacheva [57] (see also [41]). Let M denote the class of groups G with ZG = MG . Question 38. For which infinite cardinals τ does the permutation group S(τ ) of a set of size τ belong to M?

933?

The answer is positive for τ = ω [23]. Question 39. For which uncountable cardinals τ does the free group of rank τ belong to M?

934?

Question 40. (i) Is M closed under taking subgroups? In particular, do all subgroups of S(ω) belong to M? (ii) Is M closed under taking (finite) direct products?

935–936?

Question 41. (i) What is the minimal cardinality of a group G such that ZG 6= MG ? (ii) Is c such a cardinality in ZFC? (iii) Is ω1 such a cardinality in ZFC?

937–939?

Let G be a group and T be any Hausdorff group topology on G. Then MG ⊆ T , and therefore the T -closure of a subset of G must be contained in its MG -closure. In other words, the MG -closure of a given set S ⊆ G is the biggest subset of G that one could possibly hope to attain by taking the closure of S in any Hausdorff group topology on G. This naturally leads to a question whether the MG -closure of S can actually be realized by taking the closure of S in some Hausdorff group topology on G. Question 42. Let G be a group of size at most 2c and E a countable family of subsets of G. Can one find a Hausdorff group topology TE on G such that the TE -closure of every E ∈ E coincides with its MG -closure? 2An Abelian group G is almost torsion-free if G[n] is finite for every n > 1.

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For an Abelian group G the answer is positive, and in fact the topology TE in this case can be chosen to be precompact [21]. The counterpart of Question 42 for ZG instead of MG has a negative (consistent) answer. Indeed, let G be an infinite group such that ZG 6= MG and MG is discrete. (The recent example of Sipacheva [57] would do.) Let e be the unit element of G. Then G \ {e} cannot be ZG -closed. Indeed, if it were, then {e} would be ZG -open, and so by homogeneity of ZG , the topology ZG would be discrete, implying ZG = MG , a contradiction. So the ZG -closure of G \ {e} must coincide with G. On the other hand, the only Hausdorff group topology T on G is the discrete topology, and so the T -closure of G \ {e} is G \ {e}. Let us consider now the counterpart of Question 42 for PG instead of MG . 941?

Question 43. Let G be a group of size at most 2c having at least one precompact Hausdorff group topology, and let E be a countable family of subsets of G. Can one find a precompact Hausdorff group topology TE on G such that the TE -closure of every E ∈ E coincides with its PG -closure? Again, for an Abelian group G the answer is positive [21]. Thereafter, we consider only Abelian groups and refer to the three equivalent topologies ZG = MG = PG as the Markov–Zariski topology, denoting it by TG . For an infinite Abelian group G, TG is neither Hausdorff, nor a group topology on G, but still has various nice properties, e.g., the space (G, TG ) is hereditarily compact, hereditarily separable and Fréchet–Urysohn, moreover it has only finitely many connected components, and each component is an irreducible space [21].

942?

Problem 44. Let G be an Abelian group with |G| ≤ 2c and E a family of subsets of G with |E| < 2|G| . Does there exist a precompact Hausdorff group topology TE on G such that the TE -closure of each E ∈ E coincides with its TG -closure? As was mentioned before, the answer is positive for countable families E [21]. Moreover, it was shown that if |G| ≤ c, one can choose the approximating topology TE to be even metric. The inequality |E| < 2|G| in the above problem is essential. Indeed, let G be an infinite Abelian group. If one takes as E the family of all subsets of G, then the existence of a Hausdorff group topology TE on G such that the TE -closure of each E ∈ E coincides with its TG -closure would obviously imply that TE = TG . Thus TG would become Hausdorff, a contradiction. The restriction on the cardinality of the group G in Questions 42, 43 and Problem 44 is obviously necessary since the closure of a countable set in a Hausdorff topology cannot exceed 2c . The problem of characterization of the potentially dense subsets S of a group G goes back to Markov [48] who proved that every infinite subset of Z is potentially dense. This was extended in [62] to Abelian groups G of size ≤ c that are either of prime exponent or almost torsion-free. Tkachenko and Yaschenko asked in [62] whether the restriction |G| ≤ c can be relaxed to |G| ≤ 2c . To clarify better the problem, let us drop all additional restrictions on the Abelian group G. Obviously,

Bohr topologies of Abelian groups

399 |S|

if S is potentially dense in G, then |G| ≤ 22 and S must be TG -dense in G. It is not clear if these two conditions are not only necessary but also sufficient for potential density. Question 45. Let G be an Abelian group and let S be an infinite subset of G |S| such that |G| ≤ 22 and S is TG -dense in G. (i) Is S potentially dense in G? (ii) Does there exist a Hausdorff precompact group topology T such that S is T -dense in G? A positive answer to both items of this question in the case of a countable set S has been given in [21] thereby providing a positive answer to the above mentioned question from [62]. 6. Bohr topologies of Abelian groups Let G be an Abelian group. Following E. van Douwen [32], we denote by G# the group G equipped with the Bohr topology, i.e., the initial topology with respect to the family of all homomorphisms of G into the circle group T. It is a well known fact, due to Glicksberg (see also [34] in this volume), that G# has no infinite compact subsets (in particular, no nontrivial convergent sequences). Therefore, G# is always sequentially complete. For future reference, we mention two fundamental properties of the Bohr topology for arbitrary Abelian groups G, H: (i) the Bohr topology of G × H coincides with the product topology of G# × H # ; (ii) if H is a subgroup of G, then H is closed in G# and its topology as a topological subgroup of G# coincides with that of H # . E. van Douwen [49] posed the following challenging problem (see also [34]): If G and H are Abelian groups of the same size, must G# and H # be homeomorphic? # A negative solution was obtained in [43] and independently, in [30]: (Vω and p) ω # (Vq ) are not homeomorphic for different primes p and q. (For every positive integer m and a cardinal κ, Vκm denotes the direct sum of κ many copies of the group Zm .) Motivated by this, let us call a pair G, H of infinite Abelian groups: (1) Bohr-homeomorphic if G# and H # are homeomorphic, (2) weakly Bohr-homeomorphic if G# can be homeomorphically embedded into H # and H # can also be homeomorphically embedded into G# . Obviously, Bohr-homeomorphic groups are weakly Bohr-homeomorphic, and the status of the converse implication is totally unclear (see Question 49(ii)). As we shall see in the sequel, weak Bohr-homeomorphism provides a more flexible tool for studying the Bohr topology than the more rigid notion of Bohr-homeomorphism, # ω # e.g, (Vω p ) and (Vq ) are not even weakly Bohr-homeomorphic for different primes p and q. If G# homeomorphically embeds into H # and H is a bounded torsion group, then G must also be a bounded torsion group [37]. In particular, boundedness is invariant under weak Bohr-homeomorphisms, i.e., if G is a bounded Abelian

943–944?

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group and the pair G, H are weakly Bohr-homeomorphic, than H must be bounded. Therefore, when studying weak Bohr-homeomorphisms (and thus Bohr-homeomorphisms), without any loss of generality whatsoever, one can consider completely separately bounded torsion Abelian groups and non-bounded Abelian groups. We start first with the class of bounded torsion Abelian groups. According to Qn Pr¨ ufer’s theorem, every infinite bounded group has the form i=1 Vκmii for certain integers mi > 0 and cardinals κi . For this reason, and in view of items (i) and (ii), the study of the Bohr topology of the bounded Abelian groups can be focused on the groups Vκm . For bounded Abelian groups G, H the following two algebraic conditions play a prominent role. (3) |mG| = |mH| whenever m ∈ N and |mG| · |mH| ≥ ω. (4) eo(G) = eo(H) and rp (G) = rp (H) for all p with rp (G) + rp (H) ≥ ω, where eo(G) is the essential order of G [9, 37], i.e., the smallest positive integer m with mG finite. Since a pair G, H satisfies (3) iff each one of these groups has a finite-index subgroup that is isomorphic to a subgroup of the other [9], we call such pairs of bounded Abelian groups G and H weakly isomorphic [9]. By (ii), weakly isomorphic bounded Abelian groups are weakly Bohr-homeomorphic. According to [9], weakly Bohr-homeomorphic bounded Abelian groups satisfy (4), i.e., weakly isomorphic ⇒ weakly Bohr-homeomorphic ⇒ (4). Let us discuss the opposite implications. For countable Abelian groups G, H the second part of (4) becomes vacuous, while eo(G) = eo(H) yields that G, H are weakly isomorphic. Analogously, one can see that (4) for groups of square-free essential order implies weak isomorphism and Bohr-homeomorphism. Hence all four properties (1)–(4) coincide for bounded Abelian groups that have square-free essential order [9, 37]. Therefore, the invariant eo(G) alone allows for a complete classification (up to Bohr-homeomorphism) of all bounded Abelian groups of this class. The situation changes completely even for the simplest uncountable bounded ω1 ω 1 Abelian groups of essential order 4. Indeed, G = Vω 4 and H = V2 × V4 are not weakly isomorphic, because ω1 = |2G| > |2H| = ω. However, we do not know whether these groups are weakly Bohr-homeomorphic: 945?

ω # 1 # 1 be homeomorphically embedded into (Vω Question 46. Can (Vω 4 ) 2 × V4 ) ?

Here is the question in the most general form: 946?

Question 47. Given a cardinal κ ≥ ω and an integer s > 1, are Vκps and Vκp ×Vω ps weakly Bohr-homeomorphic? Can this depend on p? If the answer to Question 47 is positive for all p, then bounded Abelian groups G and H would be weakly Bohr-homeomorphic if and only if (4) holds. The next question is an equivalent form of the strongest negative answer to Question 47.

Bohr topologies of Abelian groups

401

Question 48. Assume that p is a prime number, k > 1 is an integer, κ and λ are infinite cardinals such that (Vκpk )# can be homeomorphically embedded into (Vκpk−1 × Vλpk )# . Must then inequality λ ≥ κ hold?

947?

Note that a positive answer to Question 48 answers negatively Questions 46 and 47. ω ω The countable groups Vω 4 and V2 ×V4 are obviously weakly isomorphic, hence weakly Bohr-homeomorphic (see the discussion above). ω ω Question 49. (i) ([43]) Are Vω 4 and V2 × V4 Bohr-homeomorphic? (ii) Are weakly Bohr-homeomorphic bounded groups always Bohr-homeomorphic?

Question 50. Suppose that G and H are bounded Abelian groups such that G# homeomorphically embeds into H # . Does there exist a subgroup G0 of G of finite index that algebraically embeds into H?

948–949?

950?

Note that a positive answer to this question would imply, in particular, that weak Bohr-homeomorphism coincides with weak isomorphism. Hence a positive answer to this question would imply a positive answer to Question 48. Now we leave the bounded world and turn to the class of non-bounded groups. According to Hart and Kunen [40], two Abelian groups G and H are almost isomorphic if G and H have isomorphic finite index subgroups. This definition is motivated by the fact that almost isomorphic Abelian groups are always Bohrhomeomorphic [40]. The converse implication fails. Indeed, Q and Q/Z × Z are Bohr-homeomorphic [6], and yet these groups are not almost isomorphic. It is nevertheless unclear if the reverse implication holds for bounded groups. Question 51 ([43]). Are Bohr-homeomorphic bounded Abelian groups almost isomorphic?

951?

The question on whether the pairs Z, Z2 and Z, Q are Bohr-homeomorphic is raised in [5, 34]. Let us consider here the version for weak Bohr-homeomorphisms: Question 52. (i) Are Z and Q weakly Bohr-homeomorphic? (ii) Are Z and Q/Z (weakly) Bohr-homeomorphic? A positive answer to item (i) of Question 52 would yield that all torsion-free Abelian groups of a fixed finite free rank are weakly Bohr-homeomorphic. If both items have a positive answer, then the weak Bohr-homeomorphism class of Z# would comprise the class of all Abelian groups G of finite rank3 such that either G is non-torsion or G contains a copy of the group Q/Z. (In particular, all finite powers of Z, Q and Q/Z along with their finite products would become weakly Bohr-homeomorphic.) Many nice properties of Z# can be found in [44]. For a fast growing sequence an in Z# the range is a closed discrete set of Z# (see [34] for further properties of the lacunary sets in Z# ), whereas for a polynomial function n 7→ an = P (n) the range has no isolated points [44, Theorem 5.4]. Moreover, the range P (Z) is closed 3I.e., there exists n ∈ ω such that r (G) ≤ n and |G[p]| ≤ pn for every prime p. 0

952–953?

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when P (x) = xk is a monomial. For quadratic polynomials P (x) = ax2 + bx + c (a, b, c ∈ Z, a 6= 0) the situation is already more complicated: the range P (Z) is closed iff there is at most one prime that divides a, but does not divide b [44, Theorem 5.6]. This leaves open the general question. 954?

Problem 53. Characterize the polynomials P (x) ∈ Z[x] such that P (Z) is closed in Z# . Answering a question of van Douwen, Gladdines [38] found a closed countable # ω # subset of (Vω 2 ) that is not a retract of (V2 ) , while Givens [36] proved that every infinite G# contains a closed countable subset that is not a retract of G# . However, the question remains open in the case of subgroups:

955?

Question 54 (Question 81, [49]). If H is a countable subgroup of an Abelian group G, must H # be a retract of G# ? An affirmative answer to this question of E. van Douwen was obtained in [6] in the case when H is finitely generated (see also [13] for other partial results and open problems). The general case is still open. We refer the reader to [12, 14] for further information about Bohr topology. 7. Miscellanea Two nondiscrete topologies τ1 and τ2 on a set X are called transversal if τ1 ∪τ2 generates the discrete topology on X. A precompact group topology on a group does not admit a transversal group topology, and under certain natural conditions the converse is also true [27].

956?

Question 55 ([28]). Characterize locally compact groups that admit a transversal group topology. This question is resolved for locally compact Abelian groups [27] and for connected locally compact groups [28]. There exists a locally Abelian group G and a compact normal subgroup K of G such that G does not admit a transversal group topology while G/K does have a transversal group topology [27, Example 5.4]. The inverse implication remains unclear:

957?

Question 56 ([28]). If G is a topological group that admits a transversal group topology and K is a compact normal subgroup of G, does also G/K admit a transversal group topology? The answer is positive when G = K × H for some subgroup H of G [27], or when G is a locally compact Abelian group (argue as in the proof of the implication (d) ⇒ (c) of [27, Corollary 6.7]).

958–959?

Question 57 ([28]). (i) Is it true that no minimal group topology admits a transversal group topology? (ii) Does the topology of the unitary group of an infinite-dimensional Hilbert space admit a transversal group topology?

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The answer to item (i) is positive in the Abelian case. The quasi-components (respectively, the connected components) of the Abelian pseudocompact groups are precisely all (connected) precompact groups [11]. The non-Abelian case remains unclear: Problem 58 ([11]). Describe the connected components and the quasi-components of pseudocompact groups.

960?

Given a group G, let H(G) denote the family of all Hausdorff group topologies on G, and P(G) the family of all precompact Hausdorff group topologies on G. Note that H(G) and P(G) are partially ordered sets with respect to set-theoretic inclusion of topologies. Question 59. Suppose that G and H are infinite Abelian groups. Must the groups G and H be (algebraically) isomorphic (i) if the posets H(G) and H(H) are isomorphic? (ii) if the posets P(G) and P(H) are isomorphic? A relevant information (and the origin of this question) may be found in [51]. Acknowledgement. We thankfully acknowledge helpful comments on a preliminary version of this paper offered to us by M.J. Chasco, W. Comfort, K. Kunen, M. Megrelishvili, V. Pestov, E. Reznichenko and V. Uspenskij. References Arhangel0 ski˘ı,

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Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups Jorge Galindo, Salvador Hernández and Ta-Sun Wu

1. Introduction In this paper we collect some problems that have appeared in the context of harmonic analysis on locally compact groups but can be understood, and perhaps solved, adopting topological methods. Naturally, this will also produce some genuine topological questions that can be handled using methods of harmonic analysis. We start with a simple example that illustrates quite well the interplay between the two subjects. Consider the group Z of integers and let us agree to say that a sequence (nk ) ∈ Z converges to n0 when the sequence (tnk ) converges to tn0 for all t ∈ T = {t ∈ C : |t| = 1}. Are there convergent sequences under this definition? It may appear that finding some convergent sequence should not be difficult. Suppose however that {nk } is a sequence which goes to 0. Then, by hypothesis, the sequence of functions {tnk } converges pointwise to 1 on T. Or, equivalently, the sequence of functions {ei2πnk x } converges pointwise to 1 on the interval [0, 1]. Applying Lebesgue’s Dominated Convergence Theorem, it follows that the sequence R1 R1 {0} = { 0 ei2πnk x dx} converges to 0 dx = 1, which is a contradiction. Quite surprisingly we have seen that the definition of convergence given above on Z produces no nontrivial convergent sequences. This convergence actually stems from the initial topology generated by the functions n 7→ tn of Z into T. It is called the Bohr topology of Z (denoted Z] ) and is the largest precompact (and, therefore, nondiscrete) group topology that can be defined on the integers. Even though this topology has been widely studied recently, we are still far from understanding it well in general. There are other more topological approaches to show the absence of nontrivial convergent sequences in Z] . The one we shall focus on in this paper is based on a careful study of the mappings n 7→ tn of Z into T. When a sequence of integers m {mj } is lacunary, i.e., mj+1 > q > 1, the subset A = {mj : j ∈ N} lives in Z] as an j interpolation subset: that is to say, every real-valued bounded function f (regardless of its continuity) defined on A can be extended to a continuous function f of Z] into R (alternatively, we can say that the subset {mj : j ∈ N} is C ∗ -embedded in Z] ). It is easily verified that a convergent sequence cannot be an interpolation set

The two first named authors acknowledge partial financial support by the Spanish Ministry of Science (including FEDER funds), grant MTM2004-07665-C02-01; and Fundaci´ o Caixa Castell´ o (Bancaja), grant P1 1B2005-22. 407

408

§42. Galindo, Hern´ andez, Wu, Chu duality and Bohr compactifications

and, since every sequence contains many lacunary subsequences (and, therefore, interpolation sets), it follows that there are no convergent sequences in Z] . This property actually extends to all abelian groups. If G is an abelian group, let us denote by G] , the group G equipped with its maximal precompact group topology and by bG the completion of G] . In [11] van Douwen initiated a detailed analysis of the topological properties of G] and, in doing so, he disclosed to general topologists a collection of questions that had by then been in consideration in harmonic analysis for at least 30 years. He in particular proved the following theorem that we take as our starting point. Theorem 1.1 ([11]). If G is an abelian group, every A ⊂ G contains a subset D with |D| = |A| that is relatively discrete and C ∗ -embedded in bG. 2. Basic definitions 2.1. On Chu duality. Chu duality, called unitary duality by Chu [5], is based on giving a certain topological and algebraic structure to the set of finite dimensional representations of a topological group G. Denote to that end by Gxn the set of all continuous n-dimensional unitary representations of G. It follows from a result of Goto [20] that the set Gxn , equipped F with the compact-open topology, is a locally compact space. The space Gx = n

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